Chapter 2.2, Problem 25P

a.

To determine

To show: The equation $\frac{dy}{dx}=\frac{y-4x}{x-y}$ can be rewritten as, $\frac{dy}{dx}=\frac{\left(\frac{y}{x}\right)-4}{1-\left(\frac{y}{x}\right)}$.

b.

To determine

To express: The equation $\frac{dy}{dx}$ in terms of x, v and $\frac{dv}{dx}$.

c.

To determine

To show: The equation $v+x\frac{dv}{dx}=\frac{v-4}{1-v}$ is obtained on replacing y and $\frac{dy}{dx}$ in equation $\frac{dy}{dx}=\frac{\left(\frac{y}{x}\right)-4}{1-\left(\frac{y}{x}\right)}$ by the expressions from part b.

d.

To determine

The solution of the equation $x\frac{dv}{dx}=\frac{v^2-4}{1-v}$.

e.

To determine

The solution of the equation $\frac{dy}{dx}=\frac{y-4x}{x-y}$ using the substitution $v=\frac{y}{x}$ in the solution in part d.

f.

To determine

To graph: The directional field and integral curves of the equation, $\frac{dy}{dx}=\frac{y-4x}{x-y}$.