Question

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

Inlyl=1/2x^2+c
Inlyl=1/2x^2-c
Inlyl=-1/2x^2-c
Inly|=-1/2x^2+c​

Answer 1

The ODE is separable.

`(dy)/(dx) = xy \iff \frac{dy}y = x\,dx`

Integrate both sides to get

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

`\boxedy`

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| = \frac12 x^2 - C \implies \frac1y (dy)/(dx) = x \implies (dy)/(dx) = xy`

so either of the first two answers are technically correct.