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David Terr
Ph.D. Math, UC Berkeley

 

Separable Differential Equations

Separable differential equations are the simplest type of differential equation. The basic form of a separable differential equation is as follows:

where f(x) and g(y) are integrable functions of one variable.

To solve Equation (1), we first multiply both sides by g(y) dx, obtaining

Finally we integrate both sides of (2), obtaining the following general solution:

where c is an arbitrary integration constant.

 

Examples:

1. Find the general solution to the following differential equation:

where k is an arbitrary constant.

 

Solution:

First we multiply both sides of (4) by dx/y, obtaining the following:

Next we integrate both sides as follows:

Evaluating each integral, we obtain the following:

where c is an arbitrary integration constant.

Exponentiating both sides, we obtain

where C = ec is an arbitrary postivie constant. This is the general solution to (4).

 

2. Find the general solution to the following differential equation:

 

Solution:

First we multiply both sides of (9) by dx/y, obtaining

Next we integrate both sides as follows:

Evaluating both integrals, we obtain

where c is an arbitrary constant. Finally, exponentiating both sides, we obtain the general solution

where C = ec is an arbitrary postivie constant.

 

3. Find the general solution to the following differential equation:

 

Solution:

First we multiply both sides of (14) by sec y dx, obtaining

Next we integrate both sides as follows:

Evaluating both integrals, we obtain

Exponentiating both sides, we obtain the following implicit general solution:

where C = ec is an arbitrary postivie constant.