Which is true for y(x) satisfying dydx=xy-1+x-y, y(0)=0:

Question

Which is true for $y(x)$ satisfying $\dfrac{dy}{dx}=xy-1+x-y$, $y(0)=0$:

Accepted Answer

$$\frac{dy}{dx}=(x-1)(y+1) \implies \ln(y+1)=\frac{x^2}{2}-x+C.$$

Using $y(0)=0$: $C=0$. At $x=1$: $\ln(y+1)=-\dfrac12 \implies y(1)=e^{-1/2}-1$.

Solutions · 1