Laplace Tranform Method:

Solution of ODE

M.M. Yovanovich

LPTODE2.MWS

Laplace Transform. Maple worksheet to solve ODEs.

> restart:

> with(inttrans):

We can introduce the short hand notation to make the results appear as they do in many texts. We next obtain the solution to a simple second-order ODE.

> alias(Y(s)=laplace(y(t),t,s)):

Laplace transform of derivatives.

> laplace(D(y)(t), t,s);

> laplace((D@@2)(y)(t), t,s);

> laplace((D@@3)(y)(t), t,s);

Observe that the transformed first, second and third dervatives require initial conditions such as , , etc. Now a second-order ODE will be solved.

Define the ODE.

> ode1:= (D@@2)(y)(t)+4*D(y)(t) + 5*y(t) = 10*exp(t);

Laplace tranform of ODE.

> step1:= laplace(ode1,t,s);

Substitute the initial conditions.

> step2:= subs(y(0)=1, D(y)(0) = 2, step1);

Obtain the solution for Y(s). Show the partial fractions form.

> step3:= solve(step2, Y(s)); step4:= convert(step3, parfrac,s);

Obtain the solution by means of inverse Laplace transform.

> sol1:= invlaplace(step3,s,t);

Plot the solution.

> plot(sol1, t=0..5);

The solution procedure is similar to that shown in most texts.