Homework for holidays-3

[Camillo]

Solve the equation:

$xu'' +u' -xu =0 $

[Kroldar]

Thanks

[gugo82]

@ Kroldar: with \$**\$ you could get a nice convolution symbol ($**$).

[Kroldar]

Let's apply Laplace (bilateral) transform

$ccL[xu''+u'-xu] = ccL[0]$

$-2sU(s)-s^2U'(s)+sU(s)+U'(s)=0$

$U'(s)=-s/(s^2-1)U(s)$

We have now to solve a separable differential equation, whose solutions should be

- $U_1(s) = gamma_1 * 1/(sqrt(s^2-1)) AA s in (1,+oo)$

- $U_2(s) = gamma_3 * 1/(sqrt(1-s^2)) AA s in (-1,1)$

- $U_3(s) = gamma_2 * 1/(sqrt(s^2-1)) AA s in (-oo,-1)$

If now we antitransform (let's take care of the domain of Laplace transform), we obtain the solutions of the initial equation:

- $u_1(x) = c_1 * e^x/(sqrt(x)$*$e^(-x)/(sqrt(x)) AA x in (0,+oo)$

- $u_2(x) = c_2 * e^x/(sqrt(-x)$*$e^(-x)/(sqrt(-x)) AA x in (-oo,0)$