miguelmorris1997
miguelmorris1997 3d ago โ€ข 0 views

Test questions on solving first-order linear ODEs using integrating factors.

Hey there! ๐Ÿ‘‹ Solving first-order linear ODEs can seem tricky, but with the right approach, it becomes much easier! This study guide and quiz will help you master the integrating factor method. Let's dive in! ๐Ÿงฎ
๐Ÿงฎ Mathematics

1 Answers

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lauren.garza Jan 2, 2026

๐Ÿ“š Quick Study Guide

  • ๐Ÿ”ข A first-order linear ODE has the form $\frac{dy}{dx} + P(x)y = Q(x)$.
  • ๐Ÿ” The integrating factor is given by $\mu(x) = e^{\int P(x) dx}$.
  • ๐Ÿ“ Multiply the entire ODE by the integrating factor: $\mu(x)\frac{dy}{dx} + \mu(x)P(x)y = \mu(x)Q(x)$.
  • ๐Ÿงช The left side becomes the derivative of the product $\frac{d}{dx}(\mu(x)y)$, so $\frac{d}{dx}(\mu(x)y) = \mu(x)Q(x)$.
  • ๐Ÿ’ก Integrate both sides with respect to $x$: $\int \frac{d}{dx}(\mu(x)y) dx = \int \mu(x)Q(x) dx$, which simplifies to $\mu(x)y = \int \mu(x)Q(x) dx$.
  • โœ… Solve for $y$: $y = \frac{1}{\mu(x)} \int \mu(x)Q(x) dx$.

Practice Quiz

  1. What is the integrating factor for the ODE $\frac{dy}{dx} + 2y = e^{-x}$?
    1. $e^{x}$
    2. $e^{2x}$
    3. $e^{-x}$
    4. $e^{-2x}$
  2. Given the ODE $\frac{dy}{dx} - y = x$, what is the integrating factor?
    1. $e^{x}$
    2. $e^{-x}$
    3. $x$
    4. $-x$
  3. What is the solution to the ODE $\frac{dy}{dx} + y = 0$?
    1. $y = Ce^{x}$
    2. $y = Ce^{-x}$
    3. $y = C$
    4. $y = Cx$
  4. Find the integrating factor for the ODE $\frac{dy}{dx} + \frac{y}{x} = x$.
    1. $x$
    2. $\frac{1}{x}$
    3. $\ln(x)$
    4. $e^x$
  5. Solve the ODE $\frac{dy}{dx} + 3y = 6$.
    1. $y = 2 + Ce^{-3x}$
    2. $y = 2 + Ce^{3x}$
    3. $y = 6 + Ce^{-3x}$
    4. $y = Ce^{-3x}$
  6. What is the integrating factor of the differential equation $\frac{dy}{dx} + 2xy = x$?
    1. $e^{x^2}$
    2. $e^{-x^2}$
    3. $e^{x}$
    4. $e^{-x}$
  7. The integrating factor for the ODE $x\frac{dy}{dx} + y = x^2$ is:
    1. $x$
    2. $\frac{1}{x}$
    3. $e^x$
    4. $\ln{x}$
Click to see Answers
  1. B
  2. B
  3. B
  4. A
  5. A
  6. A
  7. A

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