Question

Solve the differential equation

Answer 1

\[\frac{ dy }{ dx }=\frac{ y(\ln(y)-\ln(x)+1) }{ x }\]

Answer 2

This is obviously a substitution so I said:

\[v=\frac{ y }{ x }\]

Answer 3

\(\log(y)-\log(x)=\log(y/x)\) so we have$$\frac{dy}{dx}=\frac{y}x\cdot(\log(y/x)+1)$$now let \(u=y/x\) i.e. \(y=ux\) so \(dy/dx=x\cdot du/dx+u\)

Answer 4

Awesome. I was on the right track.

Answer 5

$$x\frac{du}{dx}+u=u(1+\log u)\\x\frac{du}{dx}=u\log u\\\frac1u\log u\frac{du}{dx}=\frac1x\\\int\frac1u \log u\ du=\int\frac1x\ dx$$

Answer 6

Awesome. I was doing it fine! :P . Just forgot that Log(a/b)=log(a)-log(b) :P .

Answer 7

Thanks!

Answer 8

@oldrin.bataku y = ux
dy/dx = du/dx + u
it is not dy/dx = x * du/dx + u . Am I wrong?

Answer 9

Nope there's nothing wrong. He used product rule.

Answer 10

x is independent variable, how can we treat it as a dependent variable to apply product rule?
dy/dx respect to x, and du/dx is the same.

Answer 11

@oldrin.bataku : I think you divided wrong though.

Answer 12

Yeah but the derivative of x would just be 1 then.