AP Calculus AB 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 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: Verify if $y = e^x$ satisfies $y' = y$.

Answer: Yes, $y' = e^x = y$. Derivative of $e^x$ equals itself.

Flashcard 2: What is the general solution for $y' = 2y$?

Answer: $y = Ce^{2x}$. Exponential growth with rate constant 2.

Flashcard 3: What is the first step in verifying a solution?

Answer: Substitute the function into the differential equation. Direct substitution method for verification.

Flashcard 4: What is a particular solution?

Answer: A solution with specified initial conditions. General solution evaluated at given conditions.

Flashcard 5: What is a solution to a differential equation?

Answer: A function satisfying the differential equation. When substituted, makes the equation true.

Flashcard 6: Identify the solution form for $y'' + y = 0$.

Answer: General solution: $y = C_1 \text{cos}(x) + C_2 \text{sin}(x)$. Characteristic equation $r^2 + 1 = 0$ gives $r = \pm i$.

Flashcard 7: State the general solution for $y' = k$.

Answer: $y = kt + C$. Integrate the constant $k$ with respect to time.

Flashcard 8: What is the general solution for $y'' = 0$?

Answer: $y = C_1x + C_2$. Double integration of zero gives linear function.

Flashcard 9: Is $y = x^2 + C$ a solution to $y'' = 2$?

Answer: Yes, $y'' = 2$. Second derivative of $x^2 + C$ is constant 2.

Flashcard 10: Verify $y = x^3$ for $y'' = 6x$.

Answer: Yes, $y'' = 6x$. Second derivative of $x^3$ is $6x$.

Flashcard 11: Verify if $y = e^{3x}$ solves $y' = 3y$.

Answer: Yes, $y' = 3e^{3x}$. Derivative matches 3 times the function.

Flashcard 12: Which function is a solution to $y' = 3x^2$?

Answer: Any antiderivative of $3x^2$, e.g., $y = x^3 + C$. Integrate $3x^2$ to get $x^3 + C$.

Flashcard 13: Determine if $y = \frac{1}{x}$ solves $xy' = -y^2$.

Answer: Yes, $y' = -\frac{1}{x^2}$. Check: $x(-\frac{1}{x^2}) = -\frac{1}{x} = -(\frac{1}{x})^2$.

Flashcard 14: Identify the order of the differential equation: $y'' + 3y' - 5y = 0$.

Answer: Order 2. Highest derivative is the second, so order is 2.

Flashcard 15: What is a linear differential equation?

Answer: An equation of the form $a_n(x)y^{(n)} + ... + a_1(x)y' + a_0(x)y = g(x)$. Coefficients are functions of $x$, equation is first-degree in $y$.

Flashcard 16: Verify if $y = \frac{1}{2}e^{2x}$ solves $y' = y$.

Answer: No, $y' = e^{2x} \neq y$. $y' = e^{2x} \neq \frac{1}{2}e^{2x} = y$.

Flashcard 17: Verify $y = 2x - 1$ for $y' = 2$.

Answer: Yes, $y' = 2$. Derivative of linear function $2x - 1$ is 2.

Flashcard 18: Determine if $y = \text{sin}(x)$ solves $y'' + y = 0$.

Answer: Yes, $y'' + y = 0$. $y'' = -\sin x$, so $y'' + y = 0$.

Flashcard 19: What is a separable differential equation?

Answer: An equation where variables can be separated on opposite sides. Variables can be moved to opposite sides.

Flashcard 20: Verify if $y = Cx^{-1}$ is a solution for $xy' + y = 0$.

Answer: Yes, it satisfies $xy' + y = 0$. $y' = -Cx^{-2}$, and $xy' + y = 0$ checks out.

Flashcard 21: What is the next step after substitution in solution verification?

Answer: Check if the equation holds true. Verify both sides are equal after substitution.

Flashcard 22: Determine if $y = \text{cos}(x)$ satisfies $y'' + y = 0$.

Answer: Yes, $y'' + y = 0$. $y'' = -\cos x$, so $y'' + y = 0$.

Flashcard 23: What is the general solution for $y'' - 4y = 0$?

Answer: $y = C_1 e^{2x} + C_2 e^{-2x}$. Characteristic equation $r^2 - 4 = 0$ gives $r = \pm 2$.

Flashcard 24: Verify if $y = C \text{cos}(2x)$ solves $y'' + 4y = 0$.

Answer: Yes, it satisfies $y'' + 4y = 0$. $y'' = -4C\cos(2x)$, so $y'' + 4y = 0$.

Flashcard 25: What is the solution form for $y'' + 9y = 0$?

Answer: General solution: $y = C_1 \text{cos}(3x) + C_2 \text{sin}(3x)$. Characteristic equation $r^2 + 9 = 0$ gives $r = \pm 3i$.

Flashcard 26: Is $y = Ce^{3x}$ a solution for $y' = 3y$?

Answer: Yes, $y' = 3y$. Derivative of $Ce^{3x}$ is $3Ce^{3x} = 3y$.

Flashcard 27: Verify if $y = x^2$ satisfies $y'' = 2$.

Answer: Yes, $y'' = 2$. Second derivative of $x^2$ is constant 2.

Flashcard 28: Verify if $y = Ce^{-2x}$ solves $y' + 2y = 0$.

Answer: Yes, it satisfies $y' + 2y = 0$. $y' = -2Ce^{-2x} = -2y$, so $y' + 2y = 0$.

Flashcard 29: Determine if $y = e^{-x}$ solves $y' = -y$.

Answer: Yes, $y' = -e^{-x}$. Derivative equals negative of the function.

Flashcard 30: What is the general solution for $y' = -y$?

Answer: $y = Ce^{-x}$. Exponential decay with rate constant 1.