Please solve the following ODE using Laplace Transform::
with the condition,
I've to use:
But I couldn't get it.. So please give me solution..
Thanks in advance.
You don't need to, if you make the substitution this transforms the DE into
which is second order linear constant coefficient nonhomogeneous.
The characteristic equation is
So the homogeneous solution is .
Assume a solution of the form .
Then and .
Substituting into the DE gives
So the nonhomogeneous solution is .
The general solution is the sum of the homogeneous and nonhomogeneous solutions.
From this we know .
From the initial conditions:
From the final initial condition
So finally our final solution is