I was reading my notes and I am given 2 equations:
$(1+x)\frac{dy_0}{dx} + y_0 =0,\ \ \ \ y_0(1) =1$
$(1+x)\frac{dy_1}{dx} + y_1 = - \frac{d^2y_{0}}{dx^2} ,\ \ \ \ y_1(1) =0$
Which have the solutions:
$y_0(x) = \frac{2}{1+x}$
$y_1(x) = \frac{2}{(1+x)^3} - \frac{1}{2(1+x)}$
The problem I have is that I can't seem to get the 2 solutions. I tried solving the equations, starting with the first one, and what I got was:
$-\ln y_0 = \ln(1+x) + c$
$y_0 = -A(1+x)$ and using $y_0(1) =1$, I get $A = -\frac{1}{2}$ and so $y_0(x) = \frac{1+x}{2}$. Have I done something wrong here?
And how do I solve $(1+x)\frac{dy_1}{dx} + y_1 = - \frac{d^2y_{0}}{dx^2} ,\ \ \ \ y_1(1) =0?$ Im kind of confused with the $y_0$ and $y_1$ terms and how to approach it.