{VERSION 2 3 "SUN SPARC SOLARIS" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "2 D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 0 0 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading \+ 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE " Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 4" 5 20 1 {CSTYLE "" -1 -1 "" 1 10 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica " 1 12 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 18 "Laplace Transforms" }}{PARA 18 "" 0 "" {TEXT 260 11 "laplace.mws" }}{PARA 19 "" 0 "" {TEXT -1 36 "Douglas B. Meade\n(me ade@math.sc.edu)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "Overview" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 548 "The Laplace transform is a very common, and useful, te chnique for solving and analyzing the solution of initial value proble ms for linear and constant coefficient equations and systems. The key \+ property of the Laplace transform is that derivatives are transformed \+ into algebraic powers of the transform variable; thus, the differentia l equation is transformed into an algebraic equation. The focus of thi s worksheet is to understand the Laplace transform, and the inverse La place transform, more than the actual solution of a differential equat ion." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Objectives" }}{EXCHG {PARA 15 "" 0 "" {TEXT -1 87 "to understand the definition of the Laplace transform and the inve rse Laplace transform" }}{PARA 15 "" 0 "" {TEXT -1 128 "to appreciate \+ the general applicability of Laplace transforms to the solution of lin ear ODEs (both single equations and systems)" }}{PARA 15 "" 0 "" {TEXT -1 66 "to understand and be able to work with piecewise defined \+ functions" }}{PARA 15 "" 0 "" {TEXT -1 101 "to become familiar with se veral of the different ways in which Maple can deal with Laplace trans forms" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "New Maple Commands" }}{EXCHG {PARA 15 "" 0 "" {HYPERLNK 17 "alias" 2 "alias" "" }{TEXT -1 111 " \+ create an alternate name by which an exp ression can be referenced" }}{PARA 15 "" 0 "" {HYPERLNK 17 "completesq uare" 2 "completesquare" "" }{TEXT -1 77 " com plete the square of a polynomial of degree 2 (in " }{HYPERLNK 17 "stud ent" 2 "student" "" }{TEXT -1 9 " package)" }}{PARA 15 "" 0 "" {HYPERLNK 17 "convert( ..., parfrac, s )" 2 "convert,parfrac" "" } {TEXT -1 74 " compute the partial fraction decomposition of an expression" }}{PARA 15 "" 0 "" {HYPERLNK 17 "convert( ..., piec ewise )" 2 "convert,piecewise" "" }{TEXT -1 76 " convert f rom Heaviside functions to piecewise-defined functions" }}{PARA 15 "" 0 "" {HYPERLNK 17 "dsolve( ..., method=laplace )" 2 "dsolve" "" } {TEXT -1 58 " use Laplace transforms to solve a differential equatio n" }}{PARA 15 "" 0 "" {HYPERLNK 17 "Heaviside" 2 "Dirac" "" }{TEXT -1 73 " the Heaviside step function (z ero if " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 11 "<0, one if " } {XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 22 ">0, and undefined for " } {XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 3 "=0)" }}{PARA 15 "" 0 "" {HYPERLNK 17 "invlaplace" 2 "invlaplace" "" }{TEXT -1 76 " \+ apply the inverse Laplace transform (in " } {HYPERLNK 17 "inttrans" 2 "inttrans" "" }{TEXT -1 9 " package)" }} {PARA 15 "" 0 "" {HYPERLNK 17 "laplace" 2 "laplace" "" }{TEXT -1 73 " \+ apply the Laplace transform (i n " }{HYPERLNK 17 "inttrans" 2 "inttrans" "" }{TEXT -1 9 " package)" } }{PARA 15 "" 0 "" {HYPERLNK 17 "piecewise" 2 "piecewise" "" }{TEXT -1 75 " create a piecewise-defined ``f unction''" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Optional" }}{EXCHG {PARA 15 "" 0 "" {HYPERLNK 17 "about" 2 "about" "" }{TEXT -1 67 " \+ check assumptions for a variable" }}{PARA 15 "" 0 "" {HYPERLNK 17 "assume" 2 "assume" "" }{TEXT -1 89 " \+ provide additional information or constraints on a variable " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 24 "Preliminary Preparations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "The full implementation of the Laplace transform is available only after the " }{HYPERLNK 17 "in ttrans" 2 "inttrans" "" }{TEXT -1 19 " library is loaded." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "with( inttrans );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "L ikewise, the " }{HYPERLNK 17 "student" 2 "student" "" }{TEXT -1 87 " p ackage is needed if it is desired to complete the square on an express ion (using the " }{HYPERLNK 17 "completesquare" 2 "student,completesqu are" "" }{TEXT -1 10 " command)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( student );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "The Laplace Transform" }}{EXCHG {PARA 0 "" 0 "" {TEXT 256 13 "Definition 1:" } {TEXT -1 38 " The Laplace transform of a function " }{XPPEDIT 18 0 "f " "I\"fG6\"" }{TEXT -1 32 " is defined to be the function " } {XPPEDIT 18 0 "F" "I\"FG6\"" }{TEXT -1 31 " given by the improper inte gral" }}}{EXCHG {PARA 258 "" 0 "" {XPPEDIT 18 0 "F(s) = Int( f(t)*exp( -s*t), t=0..infinity )" "/-%\"FG6#%\"sG-%$IntG6$*&-%\"fG6#%\"tG\"\"\"- %$expG6#,$*&F&F/F.F/!\"\"F//F.;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "for all values of " }{XPPEDIT 18 0 "s" "I\"sG6\"" } {TEXT -1 32 " for which the integral exists." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Example s using the definition" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "The defi nition of the Laplace transform can be used to find the transform of" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 23 "the constant function " } {XPPEDIT 18 0 "f(t)=1" "/-%\"fG6#%\"tG\"\"\"" }}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 8 "Solution" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "L T[1] := Int( 1 * exp(-s*t), t=0..infinity ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "LT[1] = value( LT[1] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 263 "Note that th is result -- in particular, the limit term -- is not what you will fin d in a table of Laplace transforms. Note, however, that Maple has prov ided additional information that indicates additional information that might enable it to simplify this answer." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 24 "New Featu re in Release 4" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 179 "The messages r eporting the information that Maple needs to determine if the integral is convergent is new to Release 4. Earlier releases simply returned t he result with the limit." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 20 "Why is the sign of " } {XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 28 " important in this problem ?" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "s" "I\"s G6\"" }{TEXT -1 49 "<0 then the exponential grows without bound as \+ " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 17 " increases. If " } {XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 142 "=0 then the integrand sim plifies to 1. In either case, the improper integral diverges; hence, t he Laplace transform can only be defined for " }{XPPEDIT 18 0 "s" "I \"sG6\"" }{TEXT -1 3 ">0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 23 "Assumptions -- optional" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Maple provides a mechanism, know n as the " }{HYPERLNK 17 "assume facility" 2 "assume" "" }{TEXT -1 273 ", by which user's can specify additional information about a vari able. While this is beyond the scope of the current discussion, intere sted students might appreciate the following steps that can be taken o btain a cleaner answer from the definition of the Laplace transform." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "assume( s>0 );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "about( s );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "LT[1] = value( LT[1] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "This is what we would expect to see as the Laplace transform of 1." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 183 "Assumptions should always be removed at the end of a problem. The simplest way to clear all ass umptions about a name is to simply reassign the name to it's unevaluat ed state. That is," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "s := ' s';" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "about( s );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 20 "" 0 " " {TEXT -1 30 "Explanation of tilde in output" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 11 "The tilde (" }{TEXT 19 1 "~" }{TEXT -1 26 ") that appea rs after the " }{TEXT 19 1 "s" }{TEXT -1 93 " is included as an indi cation that additional information has been provided about this name. " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 26 "the exponential function " }{XPPEDIT 18 0 "f(t ) = exp(a*t)" "/-%\"fG6#%\"tG-%$expG6#*&%\"aG\"\"\"F&F," }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "LT[exp] := Int( exp(a*t) * exp(-s*t ), t=0..infinity ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "LT[exp] = va lue( LT[exp] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 167 "Once again Maple needs some addit ional information. Using the same approach as in the first example, it is not difficult to see that the above limit exists only when " } {XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 4 " > \000" }{XPPEDIT 18 0 "a" "I\"aG6\"" }{TEXT -1 95 ". Then, the exponential decays to zero in the limit and the value of the improper integral is " }{XPPEDIT 18 0 "1/ (s-a)" "*&\"\"\"\"\"\",&%\"sGF$%\"aG!\"\"F(" }{TEXT -1 14 ", as expect ed." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 28 "the trigonometric function " }{XPPEDIT 18 0 "f (t) = cos(a*t)" "/-%\"fG6#%\"tG-%$cosG6#*&%\"aG\"\"\"F&F," }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "LT[cos] := Int( cos(a*t)*exp(-s*t), t=0..infinity ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "LT[cos] = valu e( LT[cos] );" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 25 "Clarification \+ of Notation" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 198 "Note that the limi t applies to the first two terms in the above expression. Ideally, Map le should have included parentheses to clearly indicate that the limit is applied to more than the first term." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 224 "Maple does \+ not provide any hints as to the information that it needs to complete \+ the evaluation of the improper integral. However, based on the previou s two examples, it seems that -- at the least -- we will need to assum e " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 134 " > 0. In fact, under exactly this assumption, each of the first two terms decays to zero a nd the Laplace transform is the last term: " }{XPPEDIT 18 0 "s/(s^2+a ^2)" "*&%\"sG\"\"\",&*$F#\"\"#F$*$%\"aG\"\"#F$!\"\"" }{TEXT -1 1 "." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Exampl es using the " }{TEXT 19 7 "laplace" }{TEXT -1 8 " command" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 19 7 "laplace" }{TEXT -1 233 " command automatically assumes the appropriate conditions on the tran sform variable. While this does simplify the determination of Laplace \+ transforms, it is the user's responsibility to ensure that the results are used appropriately." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "The Laplace transforms of the three examples could have obtained as follows:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "laplace( 1, t, s );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "laplace( exp(a*t), t, s );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "laplace( cos(a*t), t, s );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 " The Inverse Laplace Transform" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 321 " Inverse Laplace transform works similarly, except that there is no (us eful) explicit formula. Regardless, Maple can still be quite useful fo r the computation of inverse Laplace transforms. Note that many of the functions that appear as Laplace transforms in a typical Table of Lap lace Transforms involve terms involving " }{TEXT 257 1 "i" }{TEXT -1 14 ") a power of " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 3 ", " } {TEXT 258 2 "ii" }{TEXT -1 29 ") a linear factor involving " } {XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 7 ", or " }{TEXT 259 3 "iii" }{TEXT -1 42 ") an irreducible quadratic expression in " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 190 ". The key to the successful determ ination of inverse Laplace transforms is the ability to convert an exp ression into a form that can be inverted using the Table of Laplace Tr ansforms. The " }{TEXT 19 26 "convert( ..., parfrac, s )" }{TEXT -1 7 " and " }{TEXT 19 14 "completesquare" }{TEXT -1 74 " commands wil l be seen t be particularly helpful in completing this task." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Find the \+ inverse Laplace transform of " }{XPPEDIT 18 0 "F[1] (s)= (7*s-1)/(s^3 -7*s-6)" "/-&%\"FG6#\"\"\"6#%\"sG*&,&*&\"\"(\"\"\"F)F.F.\"\"\"!\"\"F., (*$F)\"\"$F.*&\"\"(F.F)F.F0\"\"'F0F0" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "S olution" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "F[1] := (7*s-1)/( s^3-7*s-6);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "It is not difficul t to see that " }{XPPEDIT 18 0 "F[1]" "&%\"FG6#\"\"\"" }{TEXT -1 30 " has at least one real root (" }{XPPEDIT 18 0 "s=-1" "/%\"sG,$\"\"\"! \"\"" }{TEXT -1 47 "). The full partial fraction decomposition of " } {XPPEDIT 18 0 "F[1]" "&%\"FG6#\"\"\"" }{TEXT -1 5 " is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "convert( F[1], parfrac, s );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "The inverse Laplace transform of each term is easily dete rmined: " }{XPPEDIT 18 0 "exp(3*t)" "-%$expG6#*&\"\"$\"\"\"%\"tGF'" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "-3*exp(-2*t)" ",$*&\"\"$\"\"\"-%$expG 6#,$*&\"\"#F%%\"tGF%!\"\"F%F-" }{TEXT -1 8 ", and " }{XPPEDIT 18 0 " 2*exp(-t)" "*&\"\"#\"\"\"-%$expG6#,$%\"tG!\"\"F$" }{TEXT -1 71 ". The se results can be confirmed by comparing with the results of the " } {TEXT 19 10 "invlaplace" }{TEXT -1 9 " command:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 33 "f[1] := invlaplace( F[1], s, t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Find the \+ inverse Laplace transform of " }{XPPEDIT 18 0 "F[2](s) = (s-1)/(s^2-2 *s+5)" "/-&%\"FG6#\"\"#6#%\"sG*&,&F)\"\"\"\"\"\"!\"\"F,,(*$F)\"\"#F,*& \"\"#F,F)F,F.\"\"&F,F." }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Solution" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "F[2] := (s-1)/(s^2-2*s+5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "Since the denominator is an ir reducible quadratic, it will be fruitless to look for a partial fracti on decomposition of " }{XPPEDIT 18 0 "F[2]" "&%\"FG6#\"\"#" }{TEXT -1 80 ". Instead, what is needed is to ``complete the square'' in the denominator of " }{XPPEDIT 18 0 "F[2]" "&%\"FG6#\"\"#" }{TEXT -1 33 ". This is achieved by using the " }{HYPERLNK 17 "completesquare" 2 " student,completesquare" "" }{TEXT -1 9 " command:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "completesquare( F[2] );" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "The \+ presence of (" }{XPPEDIT 18 0 "s-1" ",&%\"sG\"\"\"\"\"\"!\"\"" }{TEXT -1 52 ") says that the inverse Laplace transform includes " } {XPPEDIT 18 0 "exp(t)" "-%$expG6#%\"tG" }{TEXT -1 37 ". The inverse L aplace transform of " }{XPPEDIT 18 0 "s/(s^2+4)" "*&%\"sG\"\"\",&*$F# \"\"#F$\"\"%F$!\"\"" }{TEXT -1 31 " is found in the table to be " } {XPPEDIT 18 0 "cos(2t)" "-%$cosG6#*&\"\"#\"\"\"%\"tGF'" }{TEXT -1 42 " ; thus, the inverse Laplace transform of " }{XPPEDIT 18 0 "F[2]" "&% \"FG6#\"\"#" }{TEXT -1 9 " must be " }{XPPEDIT 18 0 "f[2](t) = exp(t)* cos(2t)" "/-&%\"fG6#\"\"#6#%\"tG*&-%$expG6#F)\"\"\"-%$cosG6#*&\"\"#F.F )F.F." }{TEXT -1 55 ". As before, this answer is easily checked using Maple" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f[2] := invlaplac e( F[2], s, t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 39 "Find the inverse Laplace transform of " }{XPPEDIT 18 0 "F[3](s) = exp(-2*s)/s^3" "/-&%\"FG6#\"\"$6#%\"sG*&-%$expG6#,$*&\"\" #\"\"\"F)F1!\"\"F1*$F)\"\"$F2" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Solution " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 277 "While this example looks inno cuous enough, the exponential term presents some new twists. The relev ant entries in a table are more complicated than we have encountered i n prior examples, let's forego the hand computations and simply ask Ma ple for the inverse Laplace transform." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "F[3] := exp(-2*s)/s^3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f[3] := invlaplace( F[3], s, t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f[3] := factor( f[3] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Wha t is this function " }{TEXT 19 9 "Heaviside" }{TEXT -1 145 "? One of t he easiest ways to learn about Heaviside functions is to plot this fun ction, and attempt to understand how it relates to this function." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot( f[3], t=0..5, title=`Function with Heaviside` );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Notice that the solution appears t o be identically zero for " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 40 "<2, and is quadratic with a vertex at " }{XPPEDIT 18 0 "t" "I\"t G6\"" }{TEXT -1 9 "=2 for " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 50 ">2. The quadratic behavior is seen in the term, " }{XPPEDIT 18 0 "(t-2)^2/2" "*&,&%\"tG\"\"\"\"\"#!\"\"\"\"#\"\"#F'" }{TEXT -1 100 ", that multiplies the Heaviside function. Since this is not seen for \+ t<2, it appears as though " }{XPPEDIT 18 0 "Heaviside(t-2)" "-%*Hea visideG6#,&%\"tG\"\"\"\"\"#!\"\"" }{TEXT -1 88 " must be identically \+ zero for t<2. (How might you use Maple to test this conjecture?)" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 20 "" 0 " " {TEXT -1 20 "Update for Release 4" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "A new feature in Release 4 is the ability to convert between H eaviside functions and piecewise-defined functions (see " }{HYPERLNK 17 "?convert,piecewise" 2 "convert,piecewise" "" }{TEXT -1 24 "). In t his case, we find" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "conver t( f[3], piecewise );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "This is \+ an even clearer illustration of the properties observed from the plot. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "In conclusion, the inverse Laplace transform of " } {XPPEDIT 18 0 "F[3]" "&%\"FG6#\"\"$" }{TEXT -1 27 " is identically ze ro for " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 66 "<=2 as a result \+ of the coefficient 2 in the exponential and; for " }{XPPEDIT 18 0 "t " "I\"tG6\"" }{TEXT -1 59 ">2, the presence of the cubic power in the \+ denominator of " }{XPPEDIT 18 0 "F[3]" "&%\"FG6#\"\"$" }{TEXT -1 42 " is the source of the quadratic term in " }{XPPEDIT 18 0 "f[3]" "&% \"fG6#\"\"$" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 45 "Laplace Transforms for Initial Value Problems" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 146 "This i s the type of problem that is typically of most interest when studying Laplace transforms, particularly in a differential equations course.. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 163 "To \+ illustrate the general procedure for solving an IVP using Laplace tran sforms, let's use the Laplace transform to find the solution to the in itial-value problem:" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 5 "'' - " }{XPPEDIT 18 0 "3y" " *&\"\"$\"\"\"%\"yGF$" }{TEXT -1 4 "' + " }{XPPEDIT 18 0 "2y" "*&\"\"# \"\"\"%\"yGF$" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "cos(t) + (t^2-3)*exp( -2*t)" ",&-%$cosG6#%\"tG\"\"\"*&,&*$F&\"\"#F'\"\"$!\"\"F'-%$expG6#,$*& \"\"#F'F&F'F-F'F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0)=0" "/-%\"yG6 #\"\"!F&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 7 "'(0)=1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Solution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The IVP can be described with:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "ODE := diff( y(t), t$2 ) - 3*diff( y(t), t ) + 2*y(t) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " = cos(t) + (t^2-3) * exp (-2*t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "IC := y(0) = 0, D(y)(0) \+ = 1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "The first step is to apply the Laplace transform \+ to the differential equation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ODE_lap := laplace( ODE, t, s );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Note the way \+ in which the Laplace transform of the solution is denoted in this equa tion: " }{TEXT 19 21 "laplace( y(t), t, s )" }{TEXT -1 30 ". Note al so the presence of " }{TEXT 19 4 "y(0)" }{TEXT -1 7 " and " }{TEXT 19 7 "D(y)(0)" }{TEXT -1 156 " in the transformed equation. Since the se values are known from the initial conditions, they should be introd uced at the earliest convenience. For example:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 32 "ODE_lap2 := subs( IC, ODE_lap );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Next, it is necessary to explicitly solve for the Laplace transform of the solution:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "SOLN_lap := solve( ODE_lap2, \{ laplace( y(t), t, \+ s ) \} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "The solution can be obtained in one step \+ using " }{TEXT 19 10 "invlaplace" }{TEXT -1 125 ", but the solution is also available by inspection once the right-hand side is written in i ts partial fraction decomposition." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "SOLN_lap2 := convert( SOLN_lap, parfrac, s );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "SOLN := invlaplace( SOLN_lap 2, s, t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "Compare this solution with the solutions obtained by directly inverting the Laplace transform once it is avail able" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "invlaplace( SOLN_la p, s, t );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "or the solution obt ained by directly with " }{TEXT 2 6 "dsolve" }{TEXT -1 65 " if we tell Maple to use Laplace transforms to solve the problem." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "dsolve( \{ ODE, IC \}, y(t), method =laplace );" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 15 "Point to Ponder " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Identify the homogeneous and p articular solutions in this solution" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 57 "Use Laplace transforms to find the general solution to \+ " }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 5 "'' + " }{XPPEDIT 18 0 " 2*y" "*&\"\"#\"\"\"%\"yGF$" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "cos(2*t) " "-%$cosG6#*&\"\"#\"\"\"%\"tGF'" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Solut ion" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ODE := diff( y(t), t$ 2 ) + 2*y(t) = cos(2*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "The Laplace transform of thi s equation is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ODE_lap := laplace( ODE, t, s );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "S OLN_lap := solve( ODE_lap, \{ laplace(y(t),t,s) \} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "invlaplace( SOLN_lap, s, t );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Note that the two constants in this solution are the two \+ initial conditions at " }{XPPEDIT 18 0 "t=0" "/%\"tG\"\"!" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "As before, there are seve ral alternate methods to reach the same solution, including" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "dsolve( ODE, y(t), method=la place );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "It is particularly noteworthy to compare \+ this solution with the one returned by " }{TEXT 19 6 "dsolve" }{TEXT -1 54 " when the solution method is not explicitly specified:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dsolve( ODE, y(t) );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "This type of solution has been observed previously (see \+ " }{HYPERLNK 17 "forced.mws" 1 "forced.mws" "" }{TEXT -1 141 "). The k ey in that situation was to ask Maple to apply a variety of trigonomet ric identities to convert the solution to a more standard form:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "combine( \", trig );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 20 "" 0 " " {TEXT -1 15 "Point to Ponder" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 " What is the relationship between the initial conditions " }{TEXT 19 4 "y(0)" }{TEXT -1 7 " and " }{TEXT 19 7 "D(y)(0)" }{TEXT -1 21 " a nd the constants " }{TEXT 19 3 "_C1" }{TEXT -1 7 " and " }{TEXT 19 3 "_C2" }{TEXT -1 1 "?" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Find the general solution to the differential e quation " }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 5 "'' + " } {XPPEDIT 18 0 "2*y" "*&\"\"#\"\"\"%\"yGF$" }{TEXT -1 4 "' = " } {XPPEDIT 18 0 "3*Dirac(t-1)-Heaviside(t-Pi)*sin(t) + exp(-t)" ",(*&\" \"$\"\"\"-%&DiracG6#,&%\"tGF%\"\"\"!\"\"F%F%*&-%*HeavisideG6#,&F*F%%#P iGF,F%-%$sinG6#F*F%F,-%$expG6#,$F*F,F%" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Solution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "The same techni ques can be used to solve problems that involve piecewise-defined forc ing functions (Heaviside) and/or impulse functions (Dirac)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ODE := diff( y(t), t$2 ) + 2 * diff( y(t), t )" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 " = 3 * \+ Dirac(t-1) + Heaviside(t-Pi) * sin(t-Pi) + exp(-t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "T he Laplace transform of the equation is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ODE_lap := laplace( ODE, t, s );" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 183 "Whi le we have managed to work with the long form for the Laplace transfor m of y, the alias command can be used to provide a simpler, and more natural, name to use for this quantity." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "alias( Y(s) = laplace( y(t), t, s ) ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "ODE_lap;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "SOLN_ lap := solve( ODE_lap, \{ Y(s) \} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "SOLN := invlaplace( SOLN_lap, s, t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 227 "This solution look needlessly complicated; it can be written i n a format that clearly identifies the role of the initial data and th e different Heaviside functions that appear in the solution. The main \+ terms in the solution are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "TERMS := \{ y(0), D(y)(0), Heaviside(t-1), Heaviside(t-Pi) \};" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "SOLN := collect( op(SOLN), \+ TERMS );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "A plot of the solution with homogeneous i nitial conditions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "IC := \+ y(0)=0, D(y)(0)=0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "is also ill uminating" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "plot( subs( IC , rhs(SOLN) ), t=0..20, title=`Solution to IVP with Heaviside` );" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Notice that the solution appears t o undergo changes in form at " }{XPPEDIT 18 0 "t=1" "/%\"tG\"\"\"" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "t=Pi" "/%\"tG%#PiG" }{TEXT -1 1 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 20 " " 0 "" {TEXT -1 16 "Points to Ponder" }}{EXCHG {PARA 15 "" 0 "" {TEXT -1 128 "What does the solution look like as a piecewise-defined functi on? (Do this for both general and homogeneous initial conditions.)" }} {PARA 15 "" 0 "" {TEXT -1 52 "Is the same solution can be obtained dir ectly using " }{TEXT 19 6 "dsolve" }{TEXT -1 6 " with " }{TEXT 19 14 " method=laplace" }{TEXT -1 1 "?" }}{PARA 15 "" 0 "" {TEXT -1 9 "What if " }{TEXT 19 14 "method=laplace" }{TEXT -1 13 " is omitted?" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 45 "Laplace Transforms for Systems of Linear ODEs" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 220 "One of the great strengths of the Laplace transform is i ts applicability to both single equations and systems using exactly th e same procedure; the only change is that now the algebraic equations \+ will be a linear system." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Examp le" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Find the general solution to the first-order linear system of ODEs" }}}{EXCHG {PARA 262 "" 0 "" {XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 4 "' = " }{XPPEDIT 18 0 "y + si n(t)" ",&%\"yG\"\"\"-%$sinG6#%\"tGF$" }}{PARA 263 "" 0 "" {XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 4 "' = " }{XPPEDIT 18 0 "x + cos(t)" ",&%\" xG\"\"\"-%$cosG6#%\"tGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Solution" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 46 "To simplify notation, let's create two aliases" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "alias( X(s)=laplace(x(t),t, s), Y(s)=laplace(y(t),t,s) ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "The system and its Lapla ce transform are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "SYS := \+ \{ diff( x(t), t ) = y(t) + sin(t), diff( y(t), t ) = x(t) + cos(t) \} ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "SYS_lap := laplace( SY S, t, s );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "The solution to this (linear) system of t wo equations is found to be" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "SOLN_lap := solve( SYS_lap, \{ X(s), Y(s) \} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 " The inverse Laplace transforms should become obvious -- or, at least m ore apparent -- from the partial fraction decompositions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "SOLN_lap2 := convert( SOLN_lap, par frac, s );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "SOLN_lap2 := \+ collect( SOLN_lap2, \{ x(0), y(0) \} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Thus, the gen eral solution to this system is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "SOLN := invlaplace( SOLN_lap2, s, t );" }}}{SECT 1 {PARA 20 " " 0 "" {TEXT -1 16 "Points to Ponder" }}{EXCHG {PARA 15 "" 0 "" {TEXT -1 82 "Identify a basis of solutions to the homogeneous system and a p articular solution." }}{PARA 15 "" 0 "" {TEXT -1 54 "What other method s could be used to solve this system?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} }{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Copyright" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "laplace.mws -- a Maple worksheet for use with Laplac e transforms" }}{PARA 0 "" 0 "" {TEXT -1 73 " \+ (for use with Chapter 7 of Nagle and Saff)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "Prepared by: Douglas B. Meade (7 March 1996)\n as part of the Maple S upplement for Nagle/Saff (Addison-Wesley)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Updated to Maple V, Release 4, by \+ Douglas B. Meade (27 December, 1996)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "For more information, please contact th e author at one of following addresses:" }}{PARA 0 "" 0 "" {TEXT -1 91 " Department of Mathematics, Univ. of South Caro lina, Columbia, SC 29208`" }}{PARA 0 "" 0 "" {TEXT -1 35 " \+ (803) 777-6183" }}{PARA 0 "" 0 "" {TEXT -1 38 " \+ meade@math.sc.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 35 "(copyright, Douglas B. Meade, 1996)" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 3 4 1802 }