Test Questions on Identifying and Solving Homogeneous DEs

📚 Quick Study Guide

• 🔍 Definition: A differential equation of the form $\frac{dy}{dx} = f(x, y)$ is homogeneous if $f(tx, ty) = f(x, y)$ for all $t$. This means $f(x,y)$ can be written as a function of $\frac{y}{x}$.
• 💡 Solution Method: To solve a homogeneous DE, use the substitution $v = \frac{y}{x}$, so $y = vx$ and $\frac{dy}{dx} = v + x\frac{dv}{dx}$.
• 📝 Substitute and Separate: Substitute $v$ and $\frac{dy}{dx}$ into the original equation and separate the variables to solve for $v(x)$.
• ➕ Back-Substitute: Replace $v$ with $\frac{y}{x}$ to express the solution in terms of $x$ and $y$.
• 📏 General Form: A homogeneous differential equation can often be written in the form $M(x, y)dx + N(x, y)dy = 0$, where $M$ and $N$ are homogeneous functions of the same degree.
• ➗ Test for Homogeneity: Check if $M(tx, ty) = t^n M(x, y)$ and $N(tx, ty) = t^n N(x, y)$ for some $n$. If they are, the equation is homogeneous.

Practice Quiz

1. Question 1: Which of the following differential equations is homogeneous?
   • A: $\frac{dy}{dx} = x^2 + y$
   • B: $\frac{dy}{dx} = \frac{x^2 + y^2}{xy}$
   • C: $\frac{dy}{dx} = x + y + 1$
   • D: $\frac{dy}{dx} = \frac{x}{y} + y$
2. Question 2: What is the appropriate substitution to solve the homogeneous differential equation $\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}$?
   • A: $y = vx$
   • B: $x = vy$
   • C: $y = e^x$
   • D: $x = e^y$
3. Question 3: Given the substitution $y=vx$, what is $\frac{dy}{dx}$?
   • A: $v + x\frac{dv}{dx}$
   • B: $x + v\frac{dv}{dx}$
   • C: $v + \frac{dv}{dx}$
   • D: $x\frac{dv}{dx}$
4. Question 4: After applying the substitution and separation of variables to a homogeneous equation, what is the next step?
   • A: Integrate both sides
   • B: Differentiate both sides
   • C: Substitute back $x$ and $y$
   • D: Take the logarithm
5. Question 5: Solve $\frac{dy}{dx} = \frac{y}{x}$. What is the general solution?
   • A: $y = Cx$
   • B: $x = Cy$
   • C: $y = x + C$
   • D: $x = y + C$
6. Question 6: Which of the following functions $M(x,y)$ is homogeneous of degree 2?
   • A: $x^2 + y$
   • B: $x + y^2$
   • C: $x^2 + xy + y^2$
   • D: $x^2 + xy + y$
7. Question 7: Identify the integrating factor needed after separation of variables for $\frac{dy}{dx} = \frac{2xy}{x^2-y^2}$ after applying $y = vx$.
   • A: $x$
   • B: $\frac{1}{x}$
   • C: $x^2$
   • D: $\frac{1}{x^2}$