Originally Posted by

**noppawit** I am quite not sure about my solution to solve this by using Linear equation method.

$\displaystyle y\frac{dx}{dy}-x = 2y^2 ; y(1) = 5$

Then, to follow the form $\displaystyle \frac{dy}{dx}+P(x)y=f(x)$ I get $\displaystyle \frac{dx}{dy}-(\frac{1}{y})x=2y$ (Because I assume, x=y and y=x)

Integrating factor is $\displaystyle e^{\int\frac{1}{y}dy} = e^{ln|y|} = y$

After that, I solved:

$\displaystyle \frac{d}{dy}(xy) = 2y^2$

$\displaystyle xy = \frac{2y^3}{3}+C1$

$\displaystyle x = \frac{2y^2}{3}+C2$