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How to Write and Solve Separable Differential Equations

A separable differential equation is a differential equation where the variables can be separated to either side of the equal sign. This means that you can put all the x-terms on one side and all the y-terms on the other. It is best to collect all the y-terms on the left-hand side and all the x-terms on the right-hand side. Generally, you write separable equations on this form:

Rule

Separable Differential Equations

g(y)y = f(x) g(y)dy =f(x)dx + C

g(y)y = f(x) g(y)dy =f(x)dx + C

Example 1

Solve the differential equation 2xyy = x2

2xyy = x2| ÷ 2x yy = x2 2x = x 2 ydy = x 2dx 1 2y2 = x2 4 + C y = ±x2 2 + C

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