AP Calculus AB Flashcards: Modeling Situations With Differential Equations

What this deck covers

This deck focuses on Modeling Situations With Differential Equations, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus AB.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Which method is used to solve $y' = yx$?

Answer: Separation of variables. Variables can be separated: $\frac{dy}{y} = x dx$.

Flashcard 2: What is a homogeneous differential equation?

Answer: All terms are a function of the dependent variable and its derivatives. No external forcing term, only function and derivatives.

Flashcard 3: What is the solution to $\frac{dy}{dx} = 3x^2$ with $y(0)=4$?

Answer: $y = x^3 + 4$. Integrate $3x^2$ and apply the initial condition.

Flashcard 4: Identify the independent variable in $dy/dt = -2y$.

Answer: $t$. The variable in the denominator of the derivative.

Flashcard 5: What does it mean for a DE to be linear?

Answer: Dependent variable and its derivatives are linear. No powers or products of $y$ or its derivatives.

Flashcard 6: What is an initial condition in differential equations?

Answer: A value that specifies the solution of a DE at a point. Determines a unique solution from the general solution.

Flashcard 7: What method is used to solve $y' = xy$?

Answer: Separation of variables. Can separate variables: $\frac{dy}{y} = x dx$.

Flashcard 8: What is a differential equation?

Answer: An equation involving derivatives of a function. Relates a function to its rate of change.

Flashcard 9: What is a first-order differential equation?

Answer: An equation involving the first derivative of a function. Only involves $y'$, no higher derivatives.

Flashcard 10: Find the particular solution of $y' = 2x$ given $y(1) = 3$.

Answer: $y = x^2 + 2$. Integrate $2x$ to get $x^2 + C$, then use initial condition.

Flashcard 11: Which function satisfies $y' = 0$?

Answer: Constant functions. Zero derivative means no change, so $y$ is constant.

Flashcard 12: Identify the dependent variable in $dy/dx = x^2y$.

Answer: $y$. The variable being differentiated ($y$ depends on $x$).

Flashcard 13: State the general solution for $y' = \frac{1}{2}y$.

Answer: $y = Ce^{\frac{1}{2}x}$. Exponential solution with growth rate $k = \frac{1}{2}$.

Flashcard 14: Find the integrating factor for $y' + 2y = 3x$.

Answer: $e^{2x}$. For $y' + py = q$, integrating factor is $e^{\int p dx} = e^{2x}$.

Flashcard 15: What technique solves $y' + p(x)y = q(x)$?

Answer: Integrating factor. Standard method for linear first-order equations.

Flashcard 16: What is the solution to $y' + y = 0$ with $y(0) = 3$?

Answer: $y = 3e^{-x}$. Exponential decay with initial condition applied.

Flashcard 17: Identify the particular solution of $y' = 2y$ given $y(1) = 3$.

Answer: $y = 3e^{2x - 2}$. Apply exponential solution and initial condition at $x = 1$.

Flashcard 18: Identify the order of the differential equation $y'' + 3y' - 5y = 0$.

Answer: Second order. The highest derivative is $y''$ (second derivative).

Flashcard 19: What is the general solution of $y' = ky$?

Answer: $y = Ce^{kx}$. Exponential growth/decay solution form.

Flashcard 20: What is the order of the DE $x^2y'' + xy' + y = 0$?

Answer: Second order. Highest derivative present is the second derivative $y''$.

Flashcard 21: State the form of a separable differential equation.

Answer: $\frac{dy}{dx} = g(x)h(y)$. Variables can be separated to each side of the equation.

Flashcard 22: What is the general solution of $y' = 0$?

Answer: Constant function. No change means the function remains constant.

Flashcard 23: What is a solution to $y' = y$ with $y(0) = 2$?

Answer: $y = 2e^x$. Exponential solution $y = Ce^x$ with $C = 2$ from condition.

Flashcard 24: What is a particular solution for $y' = xy$ given $y(1) = 4$?

Answer: $y = 4e^{\frac{x^2 - 1}{2}}$. Separate variables and integrate, then apply condition.

Flashcard 25: What does it mean for a DE to be separable?

Answer: It can be written as $g(y)dy = f(x)dx$. Variables can be moved to opposite sides for integration.

Flashcard 26: What is a particular solution of a differential equation?

Answer: A solution satisfying the initial conditions. General solution with constants determined by conditions.

Flashcard 27: What is the integrating factor for $y' + 3y = 0$?

Answer: $e^{3x}$. For homogeneous equation, integrating factor is $e^{3x}$.

Flashcard 28: What is the solution to $dy/dx = 3x^2$ with $y(0) = 1$?

Answer: $y = x^3 + 1$. Integrate and apply the given initial condition.

Flashcard 29: What is the form of a linear first-order DE?

Answer: $y' + p(x)y = q(x)$. Standard form for first-order linear differential equations.

Flashcard 30: What is the solution to $y' = 3y$ with $y(0) = 5$?

Answer: $y = 5e^{3x}$. Exponential solution with growth constant 3 and initial value 5.