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