Problem

Solve the (separable) differential equation

Solution

At first, this equation may not appear separable, so we must simplify the right hand side until it is clear what to do.

Now we separate the variables:

Now we want to integrate both sides, and simplify. Recall Theorem 1 in section 2.12, which gives the derivative of a function of the form e^u using the chain rule. Here, the integral is found by reversing the chain rule and including the factor 1/2 to make the result come out right.