Laplace Transform Applied to Initial Value Problems

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Example 4. Solve the following initial value problem

, y(0) =3 , y'(0) = .

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Note: If the right-hand side of a differential equation contains a Dirac impulse or a discontinuous function, you have no choice but to use the Laplace transform method. One way of doing it is by specifying the method= laplace option under the dsolve command.

> ODE1:=diff(y(t),t$2)+2*diff(y(t),t)-3*y(t)=8*exp(-t)+Dirac(t-1/2);

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> dsolve({ODE1,y(0)=3,D(y)(0)=-5},y(t),method=laplace); sol1:=rhs(%);

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You can compare the solution to the answer in your text. (The above example coincides with your homework problem 29, page 274.) We can plot the solution to see the response of the system to the right-hand side.

> plot(sol1,t=0..1);

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Occasionally, the method=laplace option under the dsolve command will not give you a desired solution. In that case, you can try to apply the Laplace transform method to your equation, step-by-step, helping Maple along the way with commands like simplify, combine, etc.

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