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

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

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Newton's law of cooling: $dT/dt = -k(T - T_s)$ where $T_s$ is the surrounding temperature. The general solution is:Read the answer and worked solution →Solve $(x^2 - y^2)\,dx + 2xy\,dy = 0$.Read the answer and worked solution →The particular solution of $\dfrac{dy}{dx} + y\tan x = \sec x$ given $y(0) = 1$ is:Read the answer and worked solution →Order of differential equation:Read the answer and worked solution →General solution of dy/dx = k y:Read the answer and worked solution →Variable separable form:Read the answer and worked solution →For dy/dx = x, general solution:Read the answer and worked solution →Solve dy/dx = (y/x):Read the answer and worked solution →Linear DE of first order: dy/dx + P(x) y = Q(x). Solution uses:Read the answer and worked solution →Integrating factor for dy/dx + y = e^x:Read the answer and worked solution →Solve dy/dx = -y/x with y(1) = 2:Read the answer and worked solution →Order and degree of (d²y/dx²)² + y = 0:Read the answer and worked solution →Homogeneous DE dy/dx = f(y/x): solved by substitutionRead the answer and worked solution →Solve dy/dx = 1 + y²:Read the answer and worked solution →For exponential decay (radioactive): dN/dt = -λN. Solution:Read the answer and worked solution →Solve dy/dx = (x + y)/(x - y) (homogeneous, deg 0):Read the answer and worked solution →Solve dy/dx + y = sin x:Read the answer and worked solution →Solve d²y/dx² + y = 0 (SHM):Read the answer and worked solution →Solve d²y/dx² - 4 dy/dx + 4y = 0:Read the answer and worked solution →For population growth with carrying capacity (logistic): dP/dt = rP(1 - P/K), the limiting value as t → ∞:Read the answer and worked solution →For RC circuit charging through resistor R: dq/dt + q/(RC) = V/R. Time constant:Read the answer and worked solution →Solve x dy/dx + y = x²:Read the answer and worked solution →Initial value problem: dy/dx = e^x with y(0) = 2. Solution:Read the answer and worked solution →Order of DE obtained by eliminating constants A, B from y = A e^x + B e^(-x):Read the answer and worked solution →Solve dy/dx = y² (Bernoulli-like):Read the answer and worked solution →For Newton's law of cooling, body at 80°C cools in room at 20°C. After time, T = 60°C. Find k if time = 5 min:Read the answer and worked solution →For DE M(x,y) dx + N(x,y) dy = 0 to be EXACT:Read the answer and worked solution →Linear independence: two solutions y₁, y₂ of 2nd-order linear DE are independent iff Wronskian W:Read the answer and worked solution →For DE y'' + 4y = 0, general solution:Read the answer and worked solution →The order of a differential equation is the order of the:Read the answer and worked solution →

Worked practice questions

Newton's law of cooling: $dT/dt = -k(T - T_s)$ where $T_s$ is the surrounding temperature. The general solution is:

  1. $T = T_s + c e^{-kt}$
  2. $T = T_s + c e^{kt}$
  3. $T = T_s - ct$
  4. $T = T_s t + c$
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Correct answer: A. $T = T_s + c e^{-kt}$

Substituting $u = T - T_s$ gives $du/dt = -ku$ ⇒ $u = c e^{-kt}$ ⇒ $T = T_s + c e^{-kt}$. As $t \to \infty$, $T \to T_s$ (body equilibrates with surroundings).

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Solve $(x^2 - y^2)\,dx + 2xy\,dy = 0$.

  1. $x^2 + y^2 = cx$ (family of circles through origin)
  2. $x^2 - y^2 = cx$
  3. $xy = c$
  4. $x + y = c$
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Correct answer: A. $x^2 + y^2 = cx$ (family of circles through origin)

Homogeneous. Substitute $y = vx$: $(1 - v^2)\,dx + 2v(v\,dx + x\,dv) = 0$ ⇒ $(1 - v^2 + 2v^2)\,dx + 2vx\,dv = 0$ ⇒ $(1 + v^2)\,dx + 2vx\,dv = 0$ ⇒ $dx/x + 2v\,dv/(1+v^2) = 0$. **Integrating:** $\log x + \log(1+v^2) = \log c$ ⇒ $x(1 + v^2) = c$ ⇒ $x + y^2/x = c$ ⇒ $x^2 + y^2 = cx$.

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The particular solution of $\dfrac{dy}{dx} + y\tan x = \sec x$ given $y(0) = 1$ is:

  1. $y\sec x = \tan x + 1$
  2. $y\sec x = \sec x + 1$
  3. $y\sec x = \tan x - 1$
  4. $y = \sin x + \cos x$
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Correct answer: A. $y\sec x = \tan x + 1$

**Linear:** $P = \tan x$, IF = $e^{\int \tan x\,dx} = e^{\log\sec x} = \sec x$. So $y\sec x = \int \sec x \cdot \sec x\,dx + c = \tan x + c$. At $x=0, y=1$: $1\cdot 1 = 0 + c$ ⇒ $c = 1$. **Hence $y\sec x = \tan x + 1$.**

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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