Chapter 4.7, Problem 26E

(a)

To determine

Whether the solutions $y_1\left(t\right)=t^3\text{ and }{y}_2\left(t\right)=\left|t^3\right|$ are linearly independent in the interval $\left[0,\infty \right)$.

(b)

To determine

Whether the solutions $y_1\left(t\right)=t^3\text{ and }{y}_2\left(t\right)=\left|t^3\right|$ are linearly independent in the interval $\left(-\infty ,0\right]$.

(c)

To determine

Whether the solutions $y_1\left(t\right)=t^3\text{ and }{y}_2\left(t\right)=\left|t^3\right|$ are linearly independent in the interval $\left(-\infty ,\infty \right)$.

(d)

To determine

The Wronskian $W\left[y_1,y_2\right]\left(t\right)$ for $y_1\left(t\right)=t^3\text{ and }{y}_2\left(t\right)=\left|t^3\right|$ on the interval $\left(-\infty ,\infty \right)$.