Topic 7.6 Finding General Solutions Using Separation of Variables: solve a separable differential equation for the general solution.

What this topic is asking

The College Board (Topic 7.6) gives the AB technique for solving differential equations: separation of variables. When $\frac{dy}{dx}$ can be written as a product of a function of $x$ and a function of $y$, you separate the variables to opposite sides and integrate each.

The method

A worked general solution

Handling the constant correctly

A single constant of integration captures the whole family. Although both integrals technically produce a constant, you combine them into one $+C$ on the side with the $x$-integral. When the $y$-integral yields a logarithm, as in $\frac{dy}{dx} = ky$ giving $\ln|y| = kx + C_1$, you exponentiate to solve for $y$, and the constant transforms: $|y| = e^{C_1}e^{kx}$, so $y = Ce^{kx}$ with $C = \pm e^{C_1}$ now an arbitrary nonzero constant (and $C = 0$ recovers the equilibrium $y = 0$). Tracking how the constant changes through exponentiation is a frequent source of error and a frequent source of marks.

Why separation is the only AB solving tool

On AP Calculus AB, separation of variables is the only analytic method for solving differential equations; integrating factors and other techniques are out of scope. So every solvable AB differential equation is separable, and the first step is always to check that the right side factors into an $x$-part times a $y$-part. If it does not factor (for example $\frac{dy}{dx} = x + y$), AB does not ask you to solve it analytically; you analyze it with a slope field instead. Recognizing separability quickly, then separating and integrating cleanly, with one constant and an explicit solve for $y$ when asked, is the whole skill.

Recognizing separable form

A differential equation is separable when the right side can be written as a product (or quotient) of a function of $x$ and a function of $y$. So $\frac{dy}{dx} = xy^2$, $\frac{dy}{dx} = \frac{x}{y}$, and $\frac{dy}{dx} = y\cos x$ are all separable, while $\frac{dy}{dx} = x + y$ and $\frac{dy}{dx} = x^2 + y^2$ are not, because the right side is a sum that does not factor. The quick test is to try to gather all $y$'s with $dy$ on one side and all $x$'s with $dx$ on the other; if a stray mixed term blocks the separation, the equation is not separable. Making this check before attempting to integrate saves you from a dead end and tells you when to switch to a slope-field analysis instead.

Integrating each side correctly

Both sides of a separated equation are integrals you must evaluate with the Unit 6 toolkit: basic rules, rewriting, and substitution. The $y$-side often needs care, for instance $\int \frac{1}{y}\,dy = \ln|y|$ rather than a power-rule result, and $\int \frac{1}{y^2}\,dy = -\frac{1}{y}$. Errors in these antiderivatives propagate into a wrong general solution even when the separation step is correct. After integrating, combine the two constants of integration into one, conventionally placed on the $x$-side, and only then solve for $y$. Treating each side as an ordinary integration problem, with the same accuracy you would bring to a standalone integral, is what makes separation reliable.