How to Solve a Differential Equation (Separation of Variables)

Differential equations in the form dy/dx = f(y)g(x) can be solved by separation of variables. If you are stuck on a math problem like this, here's how to solve it.

Instructions

1. Get the variables on opposite sides of the equation. This is the "separation" step. EXAMPLE: if dy/dx = eʸ/x², then dy/eʸ = dx/x².

2. Integrate both sides of the equation: int{dy/eʸ} = int{dx/x²}, and so -1/eʸ = -1/x + C. C is an undetermined constant. If you have an initial condition, such as y(1) = 2, you can solve for C at the end.

3. Take the equation you obtained in step 2 and solve for y. -1/eʸ = -1/x + C is equivalent to eʸ = x/(1 - Cx), and so y = ln (x/(1 - Cx))

4. Since our initial condition is y(1) = 2, we also need to solve the equation 2 = ln(1/(1-C)), or -2 = ln(1 - C). If you simplify it using the mathematical properties of logarithms, you'll end up with C = .865

5. Yay math!