Self-assessment quiz: standard form of linear first-order equations

📚 Quick Study Guide

• 🔍 A first-order linear differential equation can be written in the form: $\frac{dy}{dx} + P(x)y = Q(x)$.
• 💡 The standard form is crucial for applying methods like integrating factors.
• 📝 The integrating factor is given by $e^{\int P(x) dx}$.
• 📈 Multiplying the entire equation by the integrating factor allows you to solve for $y$.
• 🍎 The general solution involves integrating the product of the integrating factor and $Q(x)$.
• 🧭 Remember to include the constant of integration, $C$, in your general solution.

Practice Quiz

1. Question 1: Which of the following is the standard form of a first-order linear differential equation?
   1. A. $\frac{dy}{dx} = Q(x)$
   2. B. $y' + P(x) = Q(x)$
   3. C. $\frac{dy}{dx} + P(x)y = Q(x)$
   4. D. $P(x)y = Q(x)$
2. Question 2: What is the integrating factor used for the equation $\frac{dy}{dx} + 2y = e^{-x}$?
   1. A. $e^{x}$
   2. B. $e^{-x}$
   3. C. $e^{2x}$
   4. D. $e^{-2x}$
3. Question 3: Consider the equation $\frac{dy}{dx} - y = x$. What is $P(x)$ in the standard form?
   1. A. 1
   2. B. -1
   3. C. $x$
   4. D. -x
4. Question 4: Which of the following equations is already in standard form?
   1. A. $2\frac{dy}{dx} + 4y = 8x$
   2. B. $\frac{dy}{dx} + y = \sin(x)$
   3. C. $x\frac{dy}{dx} + y = x^2$
   4. D. $\frac{d^2y}{dx^2} + \frac{dy}{dx} + y = 0$
5. Question 5: What is the integrating factor for the equation $\frac{dy}{dx} + \frac{y}{x} = x^2$?
   1. A. $x$
   2. B. $\frac{1}{x}$
   3. C. $e^x$
   4. D. $\ln(x)$
6. Question 6: If the integrating factor for a first-order linear equation is $e^{3x}$, what is $P(x)$?
   1. A. $x$
   2. B. 3
   3. C. $3x$
   4. D. $e^{3x}$
7. Question 7: Given the equation $\frac{dy}{dx} + y = 0$, what is the general solution?
   1. A. $y = Ce^x$
   2. B. $y = Ce^{-x}$
   3. C. $y = C$
   4. D. $y = Cx$