Greetings,

I came across the folllowing question on Ross,Differential Equationsand was unable to solve it:

$\displaystyle y'=-y^2+xy+1$ with one solution given as $\displaystyle f(x)=x$, you are asked to find the general solution, my solution is as follows:

$\displaystyle y=f(x)+\frac{1}{v}$

$\displaystyle \frac{d}{dx}(x+\frac{1}{v})=-(x+\frac{1}{v})^2+x(x+\frac{1}{v})+1$

$\displaystyle -\frac{1}{v^2}\frac{dv}{dx}=-\frac{x}{v}-\frac{1}{v^2}$

Writing it in the standard form leads:

$\displaystyle \frac{dv}{dx}-vx=1$

I found the integrating factor:

$\displaystyle e^{\int{-xdx}}=e^\frac{-x^2}{2}$

My final result is:

$\displaystyle v=e^\frac{x^2}{2}\int{e^\frac{-x^2}{2}dx}+ce^\frac{x^2}{2}$

Afterwards I could not evaluate the indefinite integral and this seems to be an introductory question, do I have a mistake here ?

Any help is appreciated.

Thanks