Question

how to solve the differential equation of
dy/dx=2x/3y

Answer 1

This is a first order differential equation. Just separate the variables.

Answer 2

dy/3y=2x dx

Answer 3

integral of the right side in respect to x to get rid of the dx
integral of the left side in respect to y to get rid of dy

Answer 4

\[\frac{\text d y}{\text dx}=\frac{2x}{3y}\]separate the variables
\[3y{\text d y}=2x\text d x\]now integrate both sides\[\int3y{\text d y}=\int2x\text d x\]

Answer 5

Although both answers are correct, I would just like to point out that you can't integrate both sides with respect to different constants. It's a bit nit picky, but this is more or like it: \[ \int \frac{dy}{dx} \frac{1}{3y} dx = \int 2x dx \]The dx on the LHS cancel out, and we have: \[ \int \frac{dy}{3y} = \int 2xdx \]as wished. Again, if this was confusing, disconsider. No teacher is that rigorous, I think.

Answer 6

And both should be 3ydy haha. Typo there.

Answer 7

Thank you! this really helped

Answer 8

sorry yeah typo.

Answer 9

\[∫3y\text dy=∫2x\text dx\]
\[\frac{3y^2}{2}=x^2+c\]
\[y^2=\frac{3}{2}(x^2+c)\]
\[y=\sqrt{\frac{3}{2}(x^2+c)}\]