Calculus AB · CED Unit 7: Differential Equations · 14 min read · Updated 2026-05-10
General solutions via separation of variables — AP Calculus AB
AP Calculus AB · CED Unit 7: Differential Equations · 14 min read
1. Identifying Separable First-Order Differential Equations★★☆☆☆⏱ 3 min
Before applying separation of variables, you must confirm your DE is separable. Many DEs appear non-separable at first, but can be rewritten using exponent rules (e.g. $e^{x+y} = e^x e^y$) or factoring to reveal the required product form. AP Calculus AB will only ask you to solve separable DEs with this method.
Exam tip: If your DE is written as a sum of terms, check if you can factor out a common $x$ or $y$ term from the entire sum to get a product. If not, it is not separable for AP Calculus AB purposes.
2. Step-by-Step Separation of Variables Technique★★★☆☆⏱ 4 min
A general solution is the full family of functions that satisfy a first-order DE, containing one arbitrary constant to account for all possible solutions. Once you confirm a DE is separable, follow this consistent process:
Start from the separable form $\frac{dy}{dx} = g(x)h(y)$
Separate variables: move all $y$ terms (including $dy$) to the left, and all $x$ terms (including $dx$) to the right, resulting in $\frac{1}{h(y)} dy = g(x) dx$ (valid when $h(y) \neq 0$)
Integrate both sides: integrate the left with respect to $y$, the right with respect to $x$. Add only one arbitrary constant (the difference of two constants is still one constant)
If possible, solve explicitly for $y$ to get the final general solution; leave it as an implicit solution if an explicit form is not possible
Exam tip: You can rename your arbitrary constant at any point to simplify your final solution—AP exam graders accept any valid name for the constant, as long as it is clear it is arbitrary.
3. Simplifying General Solutions with Logarithms★★★☆☆⏱ 3 min
A very common case of separable DEs (especially for exponential growth/decay and cooling problems) gives a logarithmic antiderivative on the $y$ side, which requires special simplification. For the common DE form $\frac{dy}{dx} = ky$, separating gives $\frac{1}{y} dy = k dx$, and integrating gives $\ln|y| = kx + C$.
To get an explicit solution, exponentiate both sides: $|y| = e^{kx + C} = e^C e^{kx}$. We then absorb the absolute value and the constant $e^C$ into a new arbitrary constant $K = \pm e^C$, giving $y = K e^{kx}$. Allowing $K=0$ also captures the trivial solution $y=0$ from when we divided by $y$ earlier.
Exam tip: When you have $\ln|y| = f(x) + C$, remember that $e^{f(x) + C} = e^C e^{f(x)}$, not $e^{f(x)} + C$—this is one of the most common student mistakes on the AP exam.
4. Concept Check★★☆☆☆⏱ 4 min
Common Pitfalls
Why: Students misapply exponent rules, forgetting that the exponent distributes as a product, not a sum.
Why: Students are taught to add a constant to every indefinite integral, so they add one to each side and do not combine them.
Why: Students mix up differentials after separation, forgetting which variable goes with which integral.
Why: Students confuse addition and multiplication, and incorrectly attempt to separate non-separable DEs.
Why: Students memorize the antiderivative as $\ln y$ and skip the absolute value.