Second order differential nonhomogeneous equation with gaussian term

[coyote_001]

Hello forum,

I am trying to solve a differential equation for the last four hours and I can't figure out how...

here it is

$$\frac{d^2x(t)}{dt^2} + \frac{dx(t)}{dt} + c x(t) = d e^{-a t^2}$$

actually my problem is how to handle the Gaussian term...



if anyone can help please post...

and also i have another one (but its not so important),

$$\frac{d^2x(t)}{dt^2} + \frac{dx(t)}{dt} + c x(t) + d x^2(t) = f e^{-a t^2}$$

thanks in advance

 

[betel]

What techniques have you used to solve the equations?

The second is a completely different type of equation b/c of the nonlinear term.

 

[coyote_001]

I didn't solve the equation yet.

The second equation is nonlinear... From the nonlinear equation through some perturbations I concluded to the first linear non-homogeneous equation...

But still no solution... I am trying different methods but no solution...
I know that is solvable because mathematica returns a specific answer (huge one) including the ErrorFunction...

 

[betel]

Sorry for the mistake. I meant what techniques have you tried to solve the equation.

Doing variation of constants twice will work.

 

[vela]

You could also try using Fourier or Laplace transforms.

 

[jackmell]

Use reduction of order. Find a DE textbook on the subject and work through the example first. Here's the general outline. Now try and work through each step with the actual problem. If you can't integrate something, just leave it in it's unevaluated integral form like I did.


First consider the homogeneous case:

$$x''+x'+cx=0$$

giving:

$$x_c=x_1(t)+x_2(t)$$

Now, let

$$x=x_1 v$$

and you should get:

$$x_1 v''+(2x_1'+x_1)v'=de^{-at^2}=R$$

Now let v'=w and obtain:

$$x_1 w'+(2x_1'+x_1) w=R$$

That's first order. Find the integrating factor and call it [itex]\mu[/tex]

Then:

$$d(\mu w)=R\mu dt$$

Integrate:

$$w=\frac{1}{\mu}\left(\int R\mu+k\right)$$

Then:

$$v=\int \frac{1}{\mu}(\int R\mu+k)$$

Finally:

$$x=x_1\int\frac{1}{\mu}\int(R\mu+k)$$