USC, Department of Mathematics, Borislav Karaivanov

MATH 242, Section 003, Elementary Differential Equations
Syllabus
Worksheet 1
Worksheet 2
Laplace transform formulas (from the cover of the textbook)
Additional Laplace transform formulas
All scores currently on record.
Solutions of the test problems: Test 1, Make-up test , Test 2.

Exercises and assignments

August 21, Thursday: In class. Syllabus distributed; Ordinary differential equations (ODE); Order of an ODE; Linear and nonlinear ODEs; Normal form; Solution of an ODE; Family of solutions; Singular solution; Example 1.
Read. section 1.1
Exercise. 1.1: 1-13 (odd numbers only),17,19,21,25.
Homework. 1.1: 2,6,12,26(a); due August 26, Tuesday.
August 26, Tuesday: In class. First- and second-order initial value problems (IVP) - definition and examples; Existence and uniqueness of a solution of IVP - theorem 1.1 and examples.
Read. section 1.2;
Exercise. 1.2: 1,3,5,7,9,11,13,17,27;
Homework. 1.2: 2,10,18; 1.3: 14; due September 2, Tuesday.
August 28, Thursday: In class. Differential equations as mathematical models; Examples: population dynamics, radioactive decay, Newton's low of cooling, spread of disease, series circuits, falling bodies, viscous damping;
Read. 1.3 all the way to the paragraph about slipping chains;
Exercise. 1.3: 5,7,13,15;
September 2, Tuesday: In class. 2.1: direction field - construction and interpretation; critical points; autonomous differential equations - construction of phase portrait and solution curves, attractors and repellers; 2.2: solution by integration; differential equations with separable variables and a method for solving them.
Read. 2.1 and 2.2;
Exercise. 2.1: 1,3,15,17,19,21,23; 2.2: 1,3,9,13,15,17,19,21,23,25,29(a);
Homework. 2.1: 18; 2.2: 2,8,12,22,24; due September 9, Tuesday.
September 4, Thursday: In class. 2.2: more examples of solving differential equations with separable variables; losing solutions; 2.3: first-order linear DE; standard form, homogeneous and non-homogeneous equations; relating the general solutions of the homogeneous and non-homogeneous equations through a particular solution of the non-homogeneous equation.
Read. 2.3 up to "The Procedure";
Exercise. 2.2: 1,3,9,13,15,17,19,21,23,25,29(a);
September 9, Tuesday: In class. Solutions of problems #2 and #22 from the homework; Procedure for solving first-order linear differential equations - theoretical aspects and examples.
Read. 2.3 up to "Functions defined by integrals" and the remarks at the end of the section;
Exercise. 2.3: 1,5,7,9,11,13,17,19,23,25,29,33;
Homework. 2.3: 6,10,16,26,34; 2.4: 2,18; due September 16, Tuesday.
September 11, Thursday: In class. #13 from 2.3; Exact equations - necessary and sufficient conditions for exactness; Method of solving exact DE; Examples.
Read. 2.4;
Exercise. 2.4: 3,5,9,13,15,19,21,25,27,29,31,33,35,37;
September 16, Tuesday: In class. #34 from 2.3; #13 from 2.4; 2.4:Integrating factor - making non-exact differential equations exact; Example; 2.5: Solving differential equations by substitution - homogeneous equations of certain order; Example.
Read. 2.5;
Exercise. 2.5: 1,3,5,7,9,13,15,17,21,23,29;
Homework. 2.4: 32,38; 2.5: 2,8,20,30; due September 23, Tuesday.
September 18, Thursday: In class. #16 from 2.3; Bernoulli equations - the method and an example; Reduction to separation of variables - the method and an example (#27 from 2.5); Modeling with first-order DE - examples (#1,2,14 from 3.1);
Read. 3.1;
Exercise. 3.1: 1,3,5,7,9,11,13,15,23,25,29.
September 23, Tuesday: In class. Review: #30 from 2.5, #21 from 2.1, #19 from 2.2, #17 from 2.3, #33 from 2.4;
Exercise. Worksheet 1 .
An additional review session will be held on Wednesday, September 26, from 7:00 pm in the regular classroom. It will be devoted to answering your questions on the material listed in the worksheet.
September 25, Thursday: In class. Test 1;
September 30, Tuesday: In class. Test overview;
Read. 3.2;
Exercise. 3.2: 1,3,13;
October 2, Thursday: In class. 4.1 Higher-order differential equations; Initial-value problems - definition, existence and uniqueness of solution; Boundary-value problems - definition, examples of non-existence and non-uniqueness of solution; Homogeneous and non-homogeneous equations, particular and general solution; 4.2 Reduction of order - finding a second solution of a linear second-order DE by reduction of order;
Read. 4.1.1, first half page of 4.1.2, 4.1.3; 4.2;
Exercise. 4.2: 1-15 (odd #'s only), 17;
Homework. 4.2: 2,10,12.
October 7, Tuesday: In class. 4.3 Second-order homogeneous linear equations with constant coefficients - distinct real roots, repeated real roots, conjugate complex roots; Higher-order homogeneous linear equations with constant coefficients - form of the general solution; Examples.
Read. 4.3;
Exercise. 4.3: 5,7,11,13,15,23,25,29,31,35,39;
Homework. 4.2: 2,10,12; 4.3: 4,12,24,34,38; 4.4: 4,10,30; due October 16, Thursday.
October 9, Thursday: In class. 4.4: Non-homogeneous equations with constant coefficients, finding a particular solution when the RHS is of certain type;
Read. 4.4;
Exercise. 4.4: 1,3,5,7,9,11,13,15,17,29,31;
Homework. 4.2: 2,10,12; 4.3: 4,12,24,34,38; 4.4: 4,10,30,38,40; due October 16, Thursday.
October 14, Tuesday: Fall break-no classes
October 16, Thursday: In class. #34 from 4.3; #10 and #40 from 4.4; 4.6: Using variation of parameters to find a particular solution of a non-homogeneous equation with an arbitrary RHS; Examples;
Read. 4.6;
Exercise. 4.6: 1(5),9,11,15(17),21,25;
Homework. 4.4: 4,10,30,38,40 (deadline extended); 4.6: 2,6,16,18; due October 21, Tuesday. Solutions of the homework problems from 4.2 and 4.3.
October 21, Tuesday: In class. #16 and #18 from 4.6; Formal criterion for deciding if two solutions (functions) are independent - Wronskian; Laplace transform - definition and examples;
Read. p.145-148; 7.1;
Exercise. p.153: 23,25,27; 7.1: 1,3,5,9,11,13,15,19,21;
Homework. 7.1: 2,14,28,32; due October 28, Tuesday. Solutions of the homework problems from 4.4 and 4.6.
October 23, Thursday: In class. 7.1 and 7.2: Laplace transform of certain functions; Inverse Laplace transform of certain functions; Laplace transform of derivatives; Laplace transform of differential equation; Examples.
Read. 7.1 and 7.2;
Exercise. 7.1: 1,3,5,9,11,13,15,19,21,23-33 (odd numbers only),37; 7.2: 1-19 (odd numbers only),29,35,39;
Homework. 7.1: 2,14,28,32; 7.2: 18,28,34; due October 28, Tuesday.
Printable table of the Laplace transform formulas.
October 28, Tuesday: In class. #14 from 7.1 and #28 from 7.2; 7.3: First translation theorem - application to finding the Laplace and the inverse Laplace transforms of functions.
Read. 7.3;
Exercise. 7.3: 1,5,7,11,13,15,17,21,29,39,41,43,45,47,49-54, 55-61 (odd numbers only),69;
Homework. 7.3: 8,18,40,48,56,70; due November 4, Tuesday.
Solutions of the homework problems from 7.1 and 7.2.
October 30, Thursday: In class. Major review of the Laplace transform and its application to solving differential equations; 7.3: Second translation theorem - application to finding the Laplace and the inverse Laplace transforms of functions.
Read. 7.3; 7.4: p.339 and example 2 only;
Exercise. 7.3: 1,5,7,11,13,15,17,21,29,39,41,43,45,47,49-54, 55-61 (odd numbers only),69; 7.4: 1,3,5;
November 4, Tuesday: In class. #56 and #69 from 7.3; 7.4: Additional properties of the Laplace transform - derivatives of Laplace transform; 7.4: 1,5,28;
Read. 7.4;
Exercise. 7.4: 1,3,5,7,9,11,13,15,17,31,35,39,43;
Homework. 7.4: 4,6,8,18,32,38; due November 11, Tuesday.
Extra homework. These problems constitute a separate set and will be graded as a regular homework assignment. They are also due November 11, Tuesday.
Solutions of the homework problems from 7.3.
Additional Laplace transform formulas.
November 6, Thursday: In class. 7.4: Laplace transform of an integral; Convolution; Laplace transform of a convolution; Integral equation; Solving integral equations using the Laplace transform; 7.5: Unit impulse and Dirac's delta function; Laplace transform of the Dirac's delta function; Solving differential equations involving the Dirac's delta function.
Read. 7.4 (up to p.344 inclusive) and 7.5;
Exercise. 7.4: 7,9,11,13,15,17,31,35,39,43; 7.5: 3,5,7,11;
Homework. Set 1: 7.4: 4,6,8,18,32,38; Set 2: Extra homework; Set 3: 7.5: 2,8,12;
Each set has to be turned in separately and will be graded as a regular homework assignment. Set 1 and Set 2 are due November 11, Tuesday. Set 3 is due November 13, Thursday.
This Worksheet 2 will help you prepare for Test 2 which is scheduled for November 13, Thursday.
November 11, Tuesday: In class. Review for Test 2;
Exercise. The problems from Worksheet 2;
Homework. Set 2 and Set 3 (from above) are due November 13, Thursday.
Solutions of the homework problems from 7.4.
Review session from 7:00 pm in rm 213 (accross from my office) in Sumwalt (not in LeConte).
November 13, Thursday: Test 2.
November 18, Tuesday: In class. Series circuits - main equation and examples; 5.1.4: #45,46,56;
Read. 5.1.4;
Exercise. 5.1: 47,53;
Homework. Set 1: 5.1: #56 with gamma equal to 10. Also, explain whether the magnitude of the charge becomes very large, or very small, or oscillates as the time passes. Due November 20, Thursday.
Set 2: 5.1: 48,52; due November 25, Tuesday.
Solutions of the homework problems from 7.5 and the extra homework set #12.
November 20, Thursday: In class. Suspended wire - setting the problem, finding the equation that describes the shape of a suspended wire, and calculating the length of a suspended wire.
Exercise. Begin reviewing the problems from Worksheet 1 and Worksheet 2.
Homework. 5.1: 48,52; due November 25, Tuesday.
Solution of the last homework problem.
Solutions of the problems from Test 2.
November 25, Tuesday: In class. Review of 2.2 and 2.3;
Exercise. Continue reviewing the problems from Worksheet 1 and Worksheet 2.
Solutions of the homework problems from 5.1.
November 27, Thursday: Thanksgiving recess - no classes.
December 2, Tuesday: In class. Review of 2.4 and 2.5;
Exercise. Continue reviewing the problems from Worksheet 1 and Worksheet 2. You can skip sections 1.2, 2.1, and 3.1 from Worksheet 1 since they are not going to be tested on the Final. Try to cover the material all the way to section 4.6 by Thursday.
All scores currently on record.
December 4, Thursday: In class. Review of 4.6 and Laplace transform (differential equations, jump function, piecewise defined functions, integral equations, convolution);
There will be a review session on Tuesday, December 9, from 7:00 pm in Sumwalt, rm 213, across from my office. Its purpose will be to answer any questions you might have and resolve any difficulties that you have encountered. Meanwhile, if you need help feel free to contact me or stop by my office.
The final exam will be held on Wednesday, December 10, from 2:00 pm in the room where the class regularly meet.