Solving Differential Equations by Separating Variables

Standard

Differential Equations are equations that involve a function and its derivatives. Sometimes they can be solved using a technique called separating variables. This only works for some differential equations.

Separable differential equations are equations of the form:
N(y)y'=M(x) which can also be written as:

N(y)\frac{dy}{dx}=M(x)

 

Example:

\frac{dy}{dx}=y^2\cos(x)

1. First we have to convert this to a separable differential equation. To do this multiply both sides by \frac{1}{y^2}:
\frac{1}{y^2} \frac{dy}{dx}=\cos(x)

2. Multiply both sides by dx:
\frac{1}{y^2}dy=\cos(x)dx

3. Integrate both sides:
-\frac{1}{y}+C_1=\sin(x)+C_2
(the C1 and C2 are arbitrary constants of integration):

4. Combine the arbitrary constants:
-\frac{1}{y}=\sin(x)+C

5. Take the negative reciprocal of both sides:
y = -\frac{1}{\sin(x)+C}

As you can see differential equations are separable if you can separate the variables and then integrate to solve.