# Separable equations

#### Fredrik373

Hi!

I have problem solving this equation

dy/dx = y/2x

The answer should be y^2 = Cx

I cannot understand why you multiply the constant with x...

#### mr fantastic

MHF Hall of Fame
Hi!

I have problem solving this equation

dy/dx = y/2x

The answer should be y^2 = Cx

I cannot understand why you multiply the constant with x...
$$\displaystyle \frac{1}{y} \, dy = \frac{1}{2} \int \frac{1}{x} \, dx$$.

Therefore $$\displaystyle \ln |y| = \frac{1}{2} \ln |x| + K = \ln (\sqrt{x}) + K$$

$$\displaystyle \Rightarrow y = \sqrt{x} e^K$$

$$\displaystyle \Rightarrow y^2 = x e^{2K} = C x$$ where $$\displaystyle C = e^{2K}$$.