Chapter 2.4, Problem 18P

(a)

To determine

To verify: That $y_1\left(t\right)=1-t$, and $y_2\left(t\right)=-\frac{t^2}{4}$ is solutions of the initial value problem $y\prime =\frac{-t+\sqrt{t^2+4y}}{2}$.

(b)

To determine

To explain: The existence of two solutions of the given problem does not contradict the uniqueness part of Theorem 2.4.2. $y_1\left(t\right)=1-t$, and $y_2\left(t\right)=-\frac{t^2}{4}$ is solutions of the initial value problem $y\prime =\frac{-t+\sqrt{t^2+4y}}{2}$.

(c)

To determine

To show: That the $y=ct-c^2$ satisfy the differential equation $y\prime =\frac{-t+\sqrt{t^2+4y}}{2}$.