Question

05* Find, for y> 0, the general solution of the differential equation dy/dx=xy.

Answer 1

The ODE is separable.

$\dfrac{dy}{dx} = xy \iff \dfrac{dy}y = x\,dx$

Integrate both sides to get

$\displaystyle \int\frac{dy}y = \int x\,dx$

$\boxed{\ln|y| = \dfrac12 x^2 + C}$

But notice that replacing the constant $C$ with $-C$ doesn't affect the solution, since its derivative would recover the same ODE as before.

$\ln|y| = \dfrac12 x^2 - C \implies \dfrac1y \dfrac{dy}{dx} = x \implies \dfrac{dy}{dx} = xy$

so either of the first two answers are technically correct.