Answer to Question #201191 in Differential Equations for Popol

Question

Find general solution of the differential equation

dy/dx=y²(1+e^x)

Answer 1

Let us find general solution of the differential equation $\frac{dy}{dx}=y^2(1+e^x).$

It is easy to see that $y=0$ is a solution of this differential equation.

Assume that $y\ne 0$. Then $\frac{dy}{y^2}=(1+e^x)dx$.

It follows that $\int\frac{dy}{y^2}=\int(1+e^x)dx$. Therefore, $-\frac{1}{y}=x+e^x+C.$

We conclude that the solution is $y=0,\ \ -\frac{1}{y}=x+e^x+C.$