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.

e -y dy = 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

y cos y dy = sin x dx

The left side may be integrated by parts

y sin y - 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