Separable equations

Objective. Separate variables and apply initial conditions.

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Learn: the key idea

A separable differential equation can be rearranged so that all terms involving y (and dy) are on one side and all terms involving x (and dx) are on the other, allowing both sides to be integrated independently. The general method writes dy/dx = f(x)g(y) as (1/g(y))dy = f(x)dx, integrates both sides, and then applies any initial condition to find the constant of integration. For example, dy/dx = xy separates to (1/y)dy = x dx, giving ln|y| = x²/2 + c.

Worked example

Solve the separable equation dy/dx = 2xy, given that y = 3 when x = 0, writing y explicitly in terms of x.

1 $Separate variables: (1/y)dy = 2x dx$
2 $Integrate both sides: ln|y| = x² + c$
3 $Use y=3 at x=0: ln 3 = 0 + c, so c = ln 3$
4 $So ln|y| = x² + ln 3, giving y = 3e^(x²)$

Answer: y = 3e^(x²)

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Common mistake. Students forget the constant of integration or apply the initial condition before fully simplifying, leading to an incorrect particular solution. Always integrate both sides fully with one constant before substituting the given condition.

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