Question

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

Answer 1

The answer is: `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`.

Now to find `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)`.