Question

I need to find the general solution for:
x^2y''+4xy'+2y=2ln(x)+5
I'm not really sure where to start. Do i divide through by the x^2?

Answer 1

Yes start by dividing by x^2

Then do a change of variable...
\[y = \frac{v}{x^2}\]
\[y' = \frac{v'}{x^2} - \frac{2v}{x^3}\]
\[y'' = \frac{v''}{x^2} - \frac{4v'}{x^3} +\frac{6v}{x^4}\]
After substitution, the equation is...
\[\frac{v''}{x^2} = 2 \ln x + 5\]
Solve for "v" by integrating twice
\[v' = \int\limits v'' dx = \int\limits x^2 (2 \ln x + 5) dx\]
\[v = \int\limits v' dx\]
** Use int by parts for above
Finally plug in "v" back into ...
\[y = \frac{v}{x^2}\]
And you have your general solution for y(x)
** Note there will be 2 constants of integration in answer