# Separable Differential Equations

 What are separable differential equations and how to solve them? This is a tutorial on solving separable differential equations of the form y ' = f(x) / g(y) Examples with detailed solutions are presented and a set of exercises is presented after the tutorials. Depending on f(x) and g(y), these equations may be solved analytically. Example 1: Solve and find a general solution to the differential equation. y ' = 3 e y x 2 Solution to Example 1: We first rewrite the given equations in differential form and with variables separated, the y's on one side and the x's on the other side as follows. e -y dy = 3 x 2 dx Integrate both side. ò e -y dy = ò 3 x 2 dx which gives -e -y + C1 = x 3 + C2 , C1 and C2 are constant of integration. We now solve the above equation for y y = - ln( - x 3 - C ) , where C = C2 - C1. As practice, verify that the solution obtained satisfy the differential equation given above. Example 2: Solve and find a general solution to the differential equation. y ' = sin x / (y cos y) Solution to Example 2: Separate variables and write in differential form. y cos y dy = sin x dx Integrate both sides ò y cos y dy = ò sin x dx The left side may be integrated by parts y sin y - ò sin y dy = - cos x y sin y + cos y + C1 = - cos x + C2 , C1 and C2 are constants of integration. In this case there is no simple formula for y as a function of x. y = (-cos x - cos y + C ) / sin y , where C = C2 - C1 Exercises: Solve the following separable differential equations. a) y ' = -9 x 2 y 2 b) y ' = - 2x e y Solutions to the above exercises a) y = 1 / (3x 3 + C) b) y = - ln (x 2 + C) More references on Differential Equations Differential Equations - Runge Kutta Method