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

📚 Quick Study Guide

• 🔢 A first-order linear ODE has the form $\frac{dy}{dx} + P(x)y = Q(x)$.
• 🔍 The integrating factor is given by $\mu(x) = e^{\int P(x) dx}$.
• 📝 Multiply the entire ODE by the integrating factor: $\mu(x)\frac{dy}{dx} + \mu(x)P(x)y = \mu(x)Q(x)$.
• 🧪 The left side becomes the derivative of the product $\frac{d}{dx}(\mu(x)y)$, so $\frac{d}{dx}(\mu(x)y) = \mu(x)Q(x)$.
• 💡 Integrate both sides with respect to $x$: $\int \frac{d}{dx}(\mu(x)y) dx = \int \mu(x)Q(x) dx$, which simplifies to $\mu(x)y = \int \mu(x)Q(x) dx$.
• ✅ 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. $\ln{x}$