Homework for holidays - 4

[Camillo]

Solve the Cauchy problem

$u''+ cu =f(x) ; x> 0 $
$u(0)=u'(0)=0 $

being $c ne 0 $.

[Paolo902]

Kroldar, many thanks also for this post.

Now I'll try to write the entire solution of the equation.

$u''+cu=f(x)$ $x>0$, $c \ne 0$

By using Laplace transform (as in the other topic) and the initial conditions Camillo has given us we get ($U(s)=ccL[u(x)](s)$):
$s^2U(s)+cU(s)=ccL[f(x)](s)$
$U=\frac{ccL[f(x)](s)}{s^2+c}$
$U=ccL[f(x)](s)*ccL((sinsqrtcx)/sqrtc)$

Finally, as I wrote above, $u(x)=f(x)**(sinsqrtcx)/sqrtc$

Thanks a lot for your help and for your post. See you!

[Kroldar]

Right!

[Paolo902]

Quite similar to the previous one. Supposing $f(x)$ Laplace-trasformable (I should have said this also before... I forgot, sorry) the solution should be ($c \ne 0$):

$u(x)=f(x)**\frac{sinsqrtcx}{sqrtc}$

If it is correct I'll send you the entire solution (now sorry but I have to go). Thank you very much indeed.

Paolo