Expansion for the maximum height

Find O( ) expansion for the time when solution is at its maximum.

> Texp := t0 + t1*epsilon + t2*epsilon^2 + O(epsilon^3);

>

This time will be a critical point of the solution.

> dHexp2 := diff( Hexp2, t ) + O(epsilon^3);

When the expansion for the time is inserted into the derivative, equations for the coefficients in the time expansion can be found:

> dHexp := series( eval( dHexp2, t=Texp ), epsilon, 3 );

> Teqn0 := op(1,dHexp) = 0;

> Teqn1 := coeff(dHexp,epsilon) = 0;

> Teqn2 := coeff(dHexp,epsilon^2) = 0;

These equations are easy to solve, but since we are already using Maple -- and will want the results for use in later computations -- let's go ahead and solve these equations:

> T0 := solve( Teqn0, {t0} );

> T1 := solve( eval(Teqn1,T0), {t1} );

> T2 := solve( eval(Teqn2,T0 union T1), {t2} );

Thus, the expansion for the time of maximum height is

> Tmax := eval( Texp, T0 union T1 union T2 );

and the corresponding expansion for the maximum height is

> Hmax := series( eval( Hexp2, t=Tmax ), epsilon );

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