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.

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- Dec 26th 2010, 12:25 AMkjchauhanODE
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. - Dec 26th 2010, 12:37 AMFernandoRevilla
- Dec 26th 2010, 12:41 AMProve It
You don't need to, if you make the substitution this transforms the DE into

which is second order linear constant coefficient nonhomogeneous.

Homogeneous solution:

The characteristic equation is

.

So the homogeneous solution is .

Nonhomogeneous solution:

Assume a solution of the form .

Then and .

Substituting into the DE gives

and

and .

So the nonhomogeneous solution is .

The general solution is the sum of the homogeneous and nonhomogeneous solutions.

So .

From this we know .

From the initial conditions:

and

and

and

.

So .

Therefore

.

From the final initial condition

.

So finally our final solution is

. - Dec 26th 2010, 01:42 AMGeneral
We do not give solutions here.

Your first step is taking the laplace of both sides.

Do it and tell us if you stuck. - Dec 26th 2010, 07:25 AMkjchauhan

http://www.mathhelpforum.com/math-he...e5956950bc.png,

with the condition, http://www.mathhelpforum.com/math-he...795ce6ab0f.png

Now stuck here.. - Dec 26th 2010, 07:31 AMGeneral
Good.

Now, simplify your fraction. - Dec 26th 2010, 08:30 AMkjchauhan
- Dec 26th 2010, 08:37 AMGeneral
Note

You will use partial fraction expansion.

But it will be too long x_x