Question

Solve the following differential equation:

(ln(x)+yx)dx=(−(ln(x)))dy

C=

Answer 1

I wonder if the ODE is supposed to be

`\left(\ln x+\frac yx\right)\,\mathrm dx=-\ln x\,\mathrm dy`

(since having
`yx` makes this look increasingly more difficult...)

Rewrite the ODE as

`\ln x(\mathrm dy)/(\mathrm dx)+\frac yx=-\ln x`

Notice that the left side is the derivative of a product:

`(\mathrm d)/(\mathrm dx)\left[y\ln x\right]=-\ln x`

Integrate both sides to get

`y\ln x=x\ln x-x+C`

`\implies\boxed{y(x)=x-\frac x{\ln x}+\frac C{\ln x}}`