Chapter 1.2, Problem 31E

To determine

(a)

To check:

Whether Theorem 1 imply the existence of a unique solution to $y\frac{dy}{dx}-4x=0$ that satisfies $y\left(x_0\right)=0$.

To determine

(b)

To show:

When $x_0\ne 0$, the equation $y\frac{dy}{dx}-4x=0$ can’t possibly have a solution in a neighborhood of $x=x_0$ that satisfies $y\left(x_0\right)=0$.

To determine

(c)

To show:

The two distinct solutions of the equation $y\frac{dy}{dx}-4x=0$ satisfying $y\left(0\right)=0$.