Hey there!
I hope I can help you out!
[tex] \frac{dy}{dx}= \frac{x^2}{y} [/tex]
Let's multiply both sides by [tex]y[/tex]
[tex]y\frac{dy}{dx}=x^{2}[/tex]
Now, multiply both sides by [tex]dx[/tex]
[tex]y*dy=x^2*dx[/tex]
Now, integrate both sides
We get the following equation after integrating
[tex] \frac{y^2}{2} = \frac{x^3}{3} +C[/tex]
Now, let's solve the value of C by plugging in y= -2 and x=3.
After doing that, you would get the value of C= -7
Now we have this equation:
[tex] \frac{y^2}{2} = \frac{x^3}{3} -7[/tex]
Multiply both sides by 2
[tex] y^2 = \frac{2x^3}{3} -14[/tex]
Now, take the square root.
[tex]y = -\sqrt{\frac{2x^3}{3} -14}[/tex]
The square root is negative because we want y to have a negative value.
That should be your answer... but there are no matching answer choices.
Hope this helps, though.
Have an awesome day! :)
