I'm puzzled over solving a DE with Laplace transform, where the coefficient are non-constant and no initial values are given. The equation is xy''+y'+xy=0.
By transforming y(x), I end up with;
x(s^2 Y(s) - sy'(0) - y(0)) + (sY(s) - y'(0)) + xY(s) = 0
.. which can be rewritten as:
Y(s)=((xs+1) y(0) + xy'(0)) / (xs^2 + s + x)
Now, since I don't have any initial values, I will simply end up with an expression based on y_0(t) and y_1(t)? I haven't been able to find any suggestions/examples for these problems in my books or internet, this is just what I derived from the theories...
If you belive this computation (I have not rigorusly justified it)
Taking the derviative with respect to s gives
again taking the dervaitive with respect to s gives
Using this your ODE becomes
I looked up the inverse laplace transform of the above
Laplace transform - Wikipedia, the free encyclopedia
it is entry #13
and is a Bessell function of the first kind of order 0.
This is good becuase this is Bessell's ODE, but I didn't get two linearly independant soltuions.
This is an interesting way to Solve this ode...