Chapter 1.2, Problem 1E

To determine

(a)

To show:

The function $ɸ\left(x\right)=x^2$ is an explicit solution to $x\frac{dy}{dx}=2y$ on the interval $\left(-\infty , \infty \right)$.

To determine

(b)

To show:

The function $ɸ\left(x\right)=e^x-x$ is an explicit solution to $\frac{dy}{dx}+y^2=e^{2x}+\left(1-2x\right)e^x+x^2-1$ on the interval $\left(-\infty , \infty \right)$.

To determine

(c)

To show:

The function $ɸ\left(x\right)=x^2-x^{-1}$ is an explicit solution to $x^2{d}^{2}y/d{x}^2=2y$ on the interval $\left(0, \infty \right)$ is shown.