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 general solution of $\frac{dy}{dx} = ky$?

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

Flashcard 2: Find the general solution for $y'' = 0$.

Answer: $y = C_1 x + C_2$. Integrating twice gives linear function with two constants.

Flashcard 3: 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 4: 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 5: 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 6: 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 7: What is the integrating factor for $y' + y = e^x$?

Answer: $e^x$. Factor needed to solve first-order linear equation.

Flashcard 8: What is the order of $y^{(4)} + 2y'' + y = 0$?

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

Flashcard 9: 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 10: 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 11: 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 12: 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 13: Is $y = x^2 - 4x$ a solution to $y'' = 2$?

Answer: Yes, it is a solution. $y' = 2x - 4$, $y'' = 2$, equation is satisfied.

Flashcard 14: Is $y = e^{3x}$ a solution to $y' = 3y$?

Answer: Yes, it is a solution. Derivative: $\frac{d}{dx}(e^{3x}) = 3e^{3x} = 3y$ ✓

Flashcard 15: What type of differential equation is $y' + 3y = 6$?

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

Flashcard 16: 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 17: 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 18: 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 19: 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 20: 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 21: 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 22: Does $y = 3x + 2$ solve $y' = 3$?

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

Flashcard 23: 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 24: What is the order of the differential equation $y''' + 2y' = 0$?

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

Flashcard 25: What is the order of the differential equation $y''' + 2y' = 0$?

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

Flashcard 26: Verify $y = \frac{1}{x}$ as a solution to $xy' + y = 0$.

Answer: Yes, it is a solution. $y' = -\frac{1}{x^2}$, so $x(-\frac{1}{x^2}) + \frac{1}{x} = 0$ ✓

Flashcard 27: Verify $y = \frac{1}{x}$ as a solution to $xy' + y = 0$.

Answer: Yes, it is a solution. $y' = -\frac{1}{x^2}$, so $x(-\frac{1}{x^2}) + \frac{1}{x} = 0$ ✓

Flashcard 28: 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 29: 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 30: What type of differential equation is $y'' + 4y = 0$?

Answer: Second-order linear homogeneous. Second-order with constant coefficients and zero right side.

Flashcard 31: 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 32: 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 33: Find the general solution for $y'' = 0$.

Answer: $y = C_1 x + C_2$. Integrating twice gives linear function with two constants.

Flashcard 34: 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 35: 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 36: 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 37: Is $y = x^2 - 4x$ a solution to $y'' = 2$?

Answer: Yes, it is a solution. $y' = 2x - 4$, $y'' = 2$, equation is satisfied.

Flashcard 38: 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 39: 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 40: 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 41: Does $y = 3x + 2$ solve $y' = 3$?

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

Flashcard 42: Verify if $y = e^{2x}$ is a solution to $\frac{dy}{dx} = 2y$.

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

Flashcard 43: What form does a separable differential equation take?

Answer: $g(y)dy = f(x)dx$. Variables can be separated to opposite sides.

Flashcard 44: 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 45: 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 46: 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 47: 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 48: What form does a separable differential equation take?

Answer: $g(y)dy = f(x)dx$. Variables can be separated to opposite sides.

Flashcard 49: 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 50: What type of differential equation is $y'' + 4y = 0$?

Answer: Second-order linear homogeneous. Second-order with constant coefficients and zero right side.

Flashcard 51: What is the integrating factor for $y' + y = e^x$?

Answer: $e^x$. Factor needed to solve first-order linear equation.

Flashcard 52: 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 53: 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 54: 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 55: 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 56: 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 57: What type of differential equation is $y' + 3y = 6$?

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

Flashcard 58: 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 59: Is $y = e^{3x}$ a solution to $y' = 3y$?

Answer: Yes, it is a solution. Derivative: $\frac{d}{dx}(e^{3x}) = 3e^{3x} = 3y$ ✓

Flashcard 60: 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 61: 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 62: What is the order of $y'' + 2xy' + y = 0$?

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

Flashcard 63: 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 64: 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 65: 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 66: 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 67: 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 68: 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 69: Verify if $y = e^{2x}$ is a solution to $\frac{dy}{dx} = 2y$.

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

Flashcard 70: What is the order of $y'' + 2xy' + y = 0$?

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

Flashcard 71: 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 72: What is the order of $y^{(4)} + 2y'' + y = 0$?

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

Flashcard 73: What is the characteristic equation for $y'' + 9y = 0$?

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

Flashcard 74: What is the characteristic equation for $y'' + 9y = 0$?

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