# How do you solve for the equation dy/dx=(3x^2)/(e^2y) that satisfies the initial condition f(0)=1/2?

La risposta è: y=1/2ln(2x^3+e).

First of all, I think ther is a mistake in your writing, I think you wanted to write:

(dy)/(dx)=(3x^2)/e^(2y).

This is a separable differential equations, so:

e^(2y)dy=3x^2dxrArrinte^(2y)dy=int3x^2dxrArr

1/2e^(2y)=x^3+c.

Ora per trovare c let's use the condition: f(0)=1/2

1/2e^(2*1/2)=0^3+crArrc=1/2e.

So the solution is:

1/2e^(2y)=x^3+1/2erArre^(2y)=2x^3+erArr2y=ln(2x^3+e)rArr

y=1/2ln(2x^3+e).