Chapter D2.1, Problem 55E

a.

To determine

To verify: The functions $y=e^t\text{ and }y=e^{-t}$ are the linearly independent solution of the differential equation $y^{″}-y=0$.

b.

To determine

To explain: The reason behind that the function $y=\sinh t\text{ and }y=\cosh t$ are linearly independent solutions of the differential equation $y^{″}-y=0$.

c.

To determine

To verify: The function $y=\sinh t\text{ and }y=\cosh t$ is a solution of the differential equation $y^{″}-y=0$.

d.

To determine

To give: Two different form of general solution of the given differential equation $y^{″}-y=0$.

e.

To determine

To verify: The function $y=e^{kt}\text{ and }y=e^{-kt}$ are linearly independent solutions of the differential equation $y^{″}-k^{2}y=0$.

f.

To determine

To express: The general solution of the differential equation $y^{″}-k^{2}y=0$ in terms of

$\left\{e^{kt},e^{-kt}\right\}\text{ and }\left\{\sinh kt,\cosh kt\right\}$.