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


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