AP Calculus AB — calculusStandard1 min read

Separable Differential Equations

Solve separable differential equations for AP Calculus AB.

Tutor AI Team2026-02-25T02:14:57.097Z

Separable differential equations can be solved by separating variables. This is the main DE-solving technique for AP Calculus AB.

Method

$\frac{dy}{dx} = f(x)g(y)$

Problem

$\frac{dy}{dx} = 2xy$ , $y(0) = 3$ .

$\frac{dy}{y} = 2x\,dx$ → $\ln|y| = x^2 + C$ → $y = Ae^{x^2}$ .

$y(0) = 3$ : $A = 3$ . Solution: $y = 3e^{x^2}$ .

Solution

Problem

$\frac{dy}{dx} = \frac{x+1}{y^2}$ , $y(0) = 1$ .

$y^2\,dy = (x+1)\,dx$ → $\frac{y^3}{3} = \frac{x^2}{2} + x + C$ .

$y(0) = 1$ : $\frac{1}{3} = C$ . So $y^3 = \frac{3x^2}{2} + 3x + 1$ .

Solution

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#ap#calculus#differential-equations#separable

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