The Laplace Transform

Solving IVP with the Laplace Transform

reset()

a:=4; b:=4; y0:=1; y1:=2; f(t):=sin(3*t);

Next we need to define the second order differential equation

ODE2:=D(D(y))(t)+a*D(y)(t)+b*y(t)-f(t)

Then we take the laplace transform of the equation

LT:=laplace(ODE2,t,s))

And solve for when LT=0

solve(LT=0,laplace(y(t),t,s),IgnoreSpecialCases))

Then we substitute in the initial conditions

ourtransform:=subs(%,[y(0)=y0,y'(0)=y1])

Then we take the inverse Laplace transform

ilaplace::addpattern(pat,u,t,res)

inverselaplace:=ilaplace(ourtransform[1],s,t)

And multiply by the Heaviside function to find the solution y(t) of the IVP

y(t):=(inverselaplace)*heaviside(t)

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