75 Points! ANY CALCULUS GENIUS OUT THERE? PLEASE HELP! Which of the following is the solution to the differential equation dy/dx=x²/y with the initial condition y(3)=-2?a. y= -2e^(-9+x^3/3)b. y= 2e^(-9+x^3/3)c. y = √2x³/3d. y = √x³/3 - 14d. y = -√x³/3 - 14

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! :)