Difficulty: Intermediate | Prerequisites: Derivative rules (power, chain), antiderivative/integration techniques, comfort with algebraic manipulation.
Differential equations show up whenever a quantity's rate of change depends on the quantity itself: population growth, radioactive decay, Newton's law of cooling, compound interest. This section introduces the idea at the most basic level. You learn two things: how to check whether a proposed function actually satisfies a given differential equation, and how to solve a narrow but important class of differential equations (separable ones) by pulling all the y's to one side and all the x's to the other. This is your first real encounter with "working backwards" from a derivative to find the original function, with a constant of integration that makes the family of solutions infinite until you pin it down with an initial condition.
A differential equation relates a function to its own derivative. You can verify a proposed solution by plugging it in and checking both sides. To solve a separable differential equation, move all y terms to one side and all x terms to the other, then integrate both sides.
Differential equation (DE)
An equation that contains a function and one or more of its derivatives. For Calc AB, you will see ordinary DEs involving dy/dx.
In simple terms, it is a statement like "the rate of change of y equals some expression involving x and y."
Solution to a differential equation
A function y = f(x) that, when substituted into the DE along with its derivative, makes the equation true for all x in the domain.
Think of it as a candidate answer. You verify it the same way you verify any equation: plug it in and see if both sides match.
Separation of variables
A technique for solving DEs of the form dy/dx = g(x) · h(y). You rewrite it so everything with y (including dy) is on one side and everything with x (including dx) is on the other, then integrate both sides independently.
Think of it as sorting: y's go left, x's go right, then you antidifferentiate each pile.
General solution
The family of all solutions, written with an arbitrary constant C. Each value of C gives one particular curve.
Particular solution
A single solution obtained by using an initial condition (a known point on the curve) to solve for C.
Given a DE and a proposed solution, the process is purely mechanical:
Compute the derivative of the proposed solution.
Substitute both the function and its derivative into the DE.
Simplify. If both sides are identical, the function is a solution. If not, it is not.
Worked example: Is y = 4x a solution to dy/dx = 4y/x?
Compute the derivative: y' = 4.
Substitute into the right side: 4y/x = 4(4x)/x = 16x/x = 16.
Compare: left side is 4, right side is 16. These are not equal, so y = 4x is not a solution.
This method works when the DE can be written as dy/dx = (expression in x) · (expression in y), or more generally when you can algebraically isolate all y-terms with dy on one side and all x-terms with dx on the other.
Steps:
Rewrite the DE so that all y-terms (including dy) are on one side and all x-terms (including dx) are on the other.
Integrate both sides independently.
Solve for y if possible. Include the constant of integration C.
Worked example: Solve y · (dy/dx) = x² · (y/y). (The lecture simplifies to dy/dx = x²/y after cancellation, giving y · dy = x² · dx.)
Separate: y dy = x² dx.
Integrate both sides: ∫ y dy = ∫ x² dx.
Result: y²/2 = x³/3 + C.
Multiply through by 2: y² = (2/3)x³ + C (the constant absorbs the factor of 2).
Solve for y: y = ±√((2/3)x³ + C).
This is the general solution. An initial condition would tell you whether to use + or – and would pin down C.
Separation of variables pattern:
dy/dx = f(x) · g(y) → (1/g(y)) dy = f(x) dx → ∫ (1/g(y)) dy = ∫ f(x) dx + C
Verification procedure:
Given DE: dy/dx = F(x, y). Proposed solution: y = h(x). Compute h'(x). Check whether h'(x) = F(x, h(x)) for all x. If yes, h is a solution.
Separable DEs model exponential growth and decay (population biology, radioactive half-life), Newton's law of cooling (forensic science and engineering thermal analysis), and simple mixing problems (chemistry, environmental science). The verification skill is the same one engineers use when checking whether a candidate formula satisfies a governing physical law.
Students sometimes verify a solution by plugging in a single point. That only checks one value of x. A solution must satisfy the DE for all x in the domain, so you need to work with the general expressions, not specific numbers.
Forgetting the constant of integration C when integrating both sides. Without C, you get a single curve rather than the full family of solutions, and you cannot satisfy an initial condition.
Treating dy/dx as though dy and dx can always be casually separated. Separation of variables works only when the DE has the right structure (the right side factors into a function of x times a function of y). Not every DE is separable.
Dropping the ± when taking a square root to solve for y. The general solution may include both the positive and negative branches; an initial condition determines which one applies.
⚠️ The AP Calc AB exam regularly includes a free-response question on separation of variables. You are expected to show every algebraic step, including the constant of integration.
⚠️ Verification questions appear as multiple-choice. They are quick points if you are careful with the algebra.
⚠️ A common exam trap is presenting a function that looks plausible but fails verification. Always substitute back in rather than guessing from the form of the expression.
⚠️ When solving by separation of variables on the exam, leaving the answer in implicit form (e.g. y² = (2/3)x³ + C) is usually acceptable unless the problem explicitly says "solve for y."
True or false: To verify a solution to a DE, you substitute a single point into both sides and check equality. (False. You substitute the general function and its derivative, then check equality for all x.)
Fill in the blank: Separation of variables requires that the right side of dy/dx can be written as ________. (A product of a function of x and a function of y, i.e. f(x) · g(y).)
True or false: The general solution to a separable DE includes an arbitrary constant C. (True.)
Fill in the blank: After separating and integrating both sides, you should include the constant of integration on ________ side(s). (One side only; the two constants from each integral combine into a single C.)
True or false: y = 4x is a solution to dy/dx = 4y/x. (False. Verification gives 4 on the left and 16 on the right.)
Q: Verify whether y = e^(3x) is a solution to dy/dx = 3y.
A: Compute y' = 3e^(3x). Substitute into the right side: 3y = 3e^(3x). Left side equals right side, so y = e^(3x) is a solution.
Q: Solve dy/dx = 2x/y by separation of variables.
A: Separate: y dy = 2x dx. Integrate: y²/2 = x² + C. Multiply by 2: y² = 2x² + C. So y = ±√(2x² + C).
Q: Why does the ± appear in the solution y = ±√((2/3)x³ + C)?
A: Because squaring y to reach y² = (2/3)x³ + C loses sign information. Both the positive and negative square roots satisfy the equation. An initial condition (a specific point the curve must pass through) determines which branch to use.
Q: A student solves a separable DE and writes y²/2 = x³/3 + C₁ and also includes + C₂ on the left. Is that necessary?
A: No. Two arbitrary constants (C₁ and C₂) can be combined into one (C = C₁ – C₂). Convention is to write a single C on one side.
Separation of variables is the gateway to slope fields and Euler's method (also in this course), where you approximate solutions graphically and numerically instead of finding them exactly. It also connects forward to Calc BC topics like logistic growth models, which are separable DEs with a specific structure. The verification technique reappears whenever you check solutions to any equation, including in linear algebra and physics.
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