Exercise 23

A first-order differential equation, $y' = f(x, y)$, is homogeneous if $f(x, y) = f(tx, ty)$.

1. Show that the equation $$\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}$$ is homogeneous.

2. Let $y = xv$ and show that the equation in part (a) can be written as $$v + x\frac{dv}{dx} = \frac{x^2 + v^2 x^2}{2v x^2}.$$ Use the fact that this new equation is separable to solve for $y$.

3. Show that any homogeneous equation $y' = f(x, y)$ can be transformed into a separable differential equation by making the substitution $y = vx$.

4. A function $f$ is said to homogeneous of degree $n$ if $f(t x, t y)= t^n f(x, y)$ for $n = 1, 2, \ldots$. Show that differential equation $$P(x, y) dx + Q(x, y) dy = 0,$$ where $P$ and $Q$ are both homogeneous of degree $n$, can be transformed into a separable differential equation using the substitution $y = vx$.

5. Solve the differential equation $$x^2 y' = 2 y^2 - x^2.$$

6. Solve the differential equation $$y' = \frac{2y^4 + x^4}{xy^3}.$$