The first step in solving a differential equation is to identify the type of ODE in the problem in order to identify a strategy for solution. In this case the given ODE is separable since we can get all of the independent variables on one side of the equation, and all of the dependent variables on the other side of the equation. We are then left with:
dy/(1 + 3y) = 6 dx/(1 + 2x)^2.
Integrating both sides with the u-substitution method gives
ln|1 + 3y| = -9/(1 + 2x) + C,
where C is some constant. We can rearrange this to solve for y, giving
y(x) = C exp[-9/(1 + 2x)] - 1/3.
We can find C using the initial condition that y(-1) = 0. Plugging in gives
y(-1) = 0 = C exp[9] - 1/3 => C = 1/(3 exp[9]),
leading to the solution to the initial value problem
y(x) = exp[-9/(1 + 2x)]/(3 exp[9]) - 1/3
