AP Calculus BC Flashcards: Verifying Solutions For Differential Equations

What this deck covers

This deck focuses on Verifying Solutions For Differential Equations, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

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: What is the integrating factor for $\frac{dy}{dx} + Py = Q$?

Answer: $e^{\textstyle\int P \, dx}$. Factor to make coefficient of $\frac{dy}{dx}$ equal to $1$.

Flashcard 2: Is $y = 2x^2 + 3$ a solution to $y' = 4x$?

Answer: Yes, it is a solution. Derivative: $\frac{dy}{dx} = 4x$ matches the equation.

Flashcard 3: Identify the particular solution for $y' = 2y$ given $y(0) = 5$.

Answer: $y = 5e^{2x}$. Using initial condition $y(0) = 5$ with $y = Ce^{2x}$.

Flashcard 4: What is the general solution for $y'' + y = 0$?

Answer: $y = C_1 \text{cos}(x) + C_2 \text{sin}(x)$. Solution using characteristic equation $r^2 + 1 = 0$.

Flashcard 5: What is the general form of a first-order differential equation?

Answer: $\frac{dy}{dx} = f(x, y)$. Standard form where $f$ depends on both $x$ and $y$.

Flashcard 6: What type of differential equation is $y' + 3y = 6$?

Answer: First-order linear. Has form $y' + Py = Q$ with constant coefficient.

Flashcard 7: What is the characteristic equation for $y'' - 4y = 0$?

Answer: $r^2 - 4 = 0$. Replace $y''$ with $r^2$ and set equal to zero.

Flashcard 8: Verify if $y = 1 + \text{ln}(x)$ satisfies $xy' = 1$.

Answer: Yes, it satisfies the equation. $y' = \frac{1}{x}$, so $x \cdot \frac{1}{x} = 1$ ✓

Flashcard 9: Verify if $y = \frac{1}{1-x}$ is a solution to $y' = y^2$.

Answer: Yes, it is a solution. $y' = \frac{1}{(1-x)^2}$ and $y^2 = \frac{1}{(1-x)^2}$ ✓

Flashcard 10: Verify if $y = \frac{1}{x}$ is a solution to $xy' + 2y = 0$.

Answer: No, it is not a solution. $y' = -\frac{1}{x^2}$, so $x(-\frac{1}{x^2}) + 2(\frac{1}{x}) = \frac{1}{x} \neq 0$

Flashcard 11: What is the form of a second-order linear homogeneous differential equation?

Answer: $a y'' + b y' + c y = 0$. Standard form with constant coefficients and zero right side.

Flashcard 12: What is the form of a non-homogeneous differential equation?

Answer: $a y'' + b y' + c y = g(x)$. Same as homogeneous but with non-zero right side.

Flashcard 13: Determine if $y = \frac{e^x}{x}$ is a solution to $x^2y' - xy = e^x$.

Answer: Yes, it is a solution. Computing $y'$ using quotient rule verifies the equation.

Flashcard 14: What is the general solution to $\frac{dy}{dx} = y$?

Answer: $y = Ce^x$, where $C$ is a constant. Solution to the simplest exponential differential equation.

Flashcard 15: What is the characteristic equation for $y'' + 3y' + 2y = 0$?

Answer: $r^2 + 3r + 2 = 0$. Replace $y''$ with $r^2$, $y'$ with $r$, and $y$ with $1$.

Flashcard 16: What is the general solution to $y'' - 5y' + 6y = 0$?

Answer: $y = C_1 e^{2x} + C_2 e^{3x}$. Using characteristic roots $r = 2, 3$ from factoring.

Flashcard 17: Determine if $y = x^3$ is a solution for $y'' = 6x$.

Answer: Yes, it is a solution. $y' = 3x^2$, $y'' = 6x$, so equation holds.

Flashcard 18: What is the order of $y'' + 2xy' + y = 0$?

Answer: Second order. Highest derivative is the second derivative.

Flashcard 19: Determine if $y = x + 1$ is a solution for $y' = 1$.

Answer: Yes, it is a solution. Derivative of $x + 1$ is constant $1$.

Flashcard 20: Verify if $y = x^3 + x$ is a solution to $y' = 3x^2 + 1$.

Answer: Yes, it is a solution. Derivative matches: $\frac{d}{dx}(x^3 + x) = 3x^2 + 1$.

Flashcard 21: Verify if $y = \text{sin}(x)$ satisfies $y'' + y = 0$.

Answer: Yes, it satisfies the equation. $y' = \cos(x)$, $y'' = -\sin(x)$, so $y'' + y = 0$ ✓

Flashcard 22: What is the order of the differential equation $y''' + 2y' = 0$?

Answer: Third order. Highest derivative is the third derivative.

Flashcard 23: Does $y = 3x + 2$ solve $y' = 3$?

Answer: Yes, it solves the equation. Derivative of $3x + 2$ is constant $3$.

Flashcard 24: Verify if $y = e^{-x}$ is a solution to $y' = -y$.

Answer: Yes, it is a solution. Derivative: $\frac{d}{dx}(e^{-x}) = -e^{-x} = -y$ ✓

Flashcard 25: What is the characteristic equation for $y'' + 9y = 0$?

Answer: $r^2 + 9 = 0$. Replace $y''$ with $r^2$ and $y$ with $1$.

Flashcard 26: What is the order of $y^{(4)} + 2y'' + y = 0$?

Answer: Fourth order. Highest derivative is the fourth derivative.

Flashcard 27: Verify if $y = \frac{x^2}{2}$ satisfies $y'' = 1$.

Answer: Yes, it satisfies the equation. $y' = x$, $y'' = 1$, so equation is satisfied.

Flashcard 28: What is the general solution of $\frac{dy}{dx} = ky$?

Answer: $y = Ce^{kx}$, where $C$ is a constant. Solution to exponential growth/decay differential equation.

Flashcard 29: Identify the particular solution for $y' = 5y$ given $y(0) = 2$.

Answer: $y = 2e^{5x}$. Using initial condition $y(0) = 2$ with $y = Ce^{5x}$.

Flashcard 30: Verify if $y = e^{2x}$ is a solution to $\frac{dy}{dx} = 2y$.

Answer: Yes, it is a solution. Taking derivative: $\frac{dy}{dx} = 2e^{2x} = 2y$ ✓