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AP Calculus AB Unit 7: Differential Equations and Modeling — practice questions — page 2

Explore 65 free Unit 7: Differential Equations and Modeling questions in AP Calculus AB Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Unit 7: Differential Equations and Modeling questions and exercises with answers

Showing questions 31–60 of 65.

The degree of a differential equation (when it is polynomial in derivatives) is the power of the:Read the answer and worked solution →The order of the differential equation $\dfrac{dy}{dx} + y = 0$ is:Read the answer and worked solution →The order and degree of $\left(\dfrac{d^2y}{dx^2}\right)^3 + \dfrac{dy}{dx} = 0$ are:Read the answer and worked solution →A first-order equation of the form $\dfrac{dy}{dx} = f(x)\,g(y)$ is solved by the method of:Read the answer and worked solution →The general solution of $\dfrac{dy}{dx} = 0$ is:Read the answer and worked solution →The number of arbitrary constants in the general solution of an $n$-th order differential equation is:Read the answer and worked solution →The general solution of $\dfrac{dy}{dx} = 2x$ is:Read the answer and worked solution →The standard form of a first-order linear differential equation is:Read the answer and worked solution →The degree of the differential equation $\left(\dfrac{dy}{dx}\right)^2 + y = 0$ is:Read the answer and worked solution →The general solution of $\dfrac{dy}{dx} = e^x$ is:Read the answer and worked solution →The order of the differential equation $\dfrac{d^2y}{dx^2} + 3\dfrac{dy}{dx} + 2y = 0$ is:Read the answer and worked solution →The **order** of a differential equation is:Read the answer and worked solution →The order and degree of $\sqrt{1 + (dy/dx)^2} = d^2y/dx^2$ are:Read the answer and worked solution →The order and degree of $x^2 \dfrac{d^2y}{dx^2} + 3x \dfrac{dy}{dx} + 4y = 0$ are respectively:Read the answer and worked solution →The **general solution** of a differential equation contains:Read the answer and worked solution →The integrating factor for the linear DE $\dfrac{dy}{dx} + P(x)y = Q(x)$ is:Read the answer and worked solution →To solve a **homogeneous** differential equation of degree 1 (e.g. $dy/dx = f(y/x)$), the standard substitution is:Read the answer and worked solution →The order of the differential equation obtained by eliminating the arbitrary constants $A$ and $B$ from $y = A e^{3x} + B e^{-3x}$ is:Read the answer and worked solution →Eliminate the arbitrary constant from $y^2 = 4ax$ to form the DE:Read the answer and worked solution →The DE obtained by eliminating $c_1$ and $c_2$ from $y = c_1 e^{3x} + c_2 e^{2x}$ is:Read the answer and worked solution →Solve $\dfrac{dy}{dx} = \dfrac{1 + y^2}{1 + x^2}$.Read the answer and worked solution →Solve $\dfrac{dy}{dx} = e^{x+y}$.Read the answer and worked solution →The DE $y - x \dfrac{dy}{dx} = 0$ has general solution:Read the answer and worked solution →The general solution of $\dfrac{dy}{dx} + y = e^{-x}$ is:Read the answer and worked solution →Solve the homogeneous DE $x\,dy/dx = x\,\tan(y/x) + y$:Read the answer and worked solution →The differential equation of the family of straight lines $y = mx + c$ (with $m, c$ arbitrary) is:Read the answer and worked solution →Solve $\sec^2 x \cdot \tan y \, dx + \sec^2 y \cdot \tan x \, dy = 0$:Read the answer and worked solution →Solve $\dfrac{dy}{dx} = \dfrac{y + \sqrt{x^2 + y^2}}{x}$.Read the answer and worked solution →The half-life of a radioactive substance decaying by $dm/dt = -km$ (with $k > 0$) in terms of $k$ is:Read the answer and worked solution →Number of rows in matrix A times number of columns in A x B should equal:Read the answer and worked solution →

Worked practice questions

The degree of a differential equation (when it is polynomial in derivatives) is the power of the:

  1. independent variable
  2. highest-order derivative
  3. lowest-order derivative
  4. dependent variable
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Correct answer: B. highest-order derivative

Degree (for a polynomial-in-derivatives equation) = power of the highest-order derivative.

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The order of the differential equation $\dfrac{dy}{dx} + y = 0$ is:

  1. 2
  2. 0
  3. 1
  4. 3
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Correct answer: C. 1

The highest derivative is dy/dx (first order), so the order is 1.

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The order and degree of $\left(\dfrac{d^2y}{dx^2}\right)^3 + \dfrac{dy}{dx} = 0$ are:

  1. order 2, degree 3
  2. order 2, degree 1
  3. order 3, degree 2
  4. order 1, degree 3
Show answer and solution

Correct answer: A. order 2, degree 3

Highest derivative is d²y/dx² (order 2), raised to power 3 (degree 3).

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More Mathematics topics

Unit 3B: Implicit Differentiation and Inverse FunctionsUnit 3A: Chain Rule and Composite FunctionsUnit 1A3: Function Composition and InversesUnit 1A1: Relations, Functions and GraphsUnit 1A2: Trigonometric Functions & Inverse TrigUnit 2: Differentiation and Fundamental Derivative RulesUnit 6A: Indefinite Integrals & Fundamental TheoremUnit 1B3: Asymptotic Behavior and Infinite Limits