Chapter 9.1, Problem 50E

a.

To determine

To verify: The function $y\left(t\right)=C_1{e}^t+C_2{e}^{-t}$ is a solution of the differential equation $y^{″}\left(t\right)-k^{2}y\left(t\right)=0$ .

b.

To determine

To verify: The given function $y\left(t\right)=C_1{e}^{2t}+C_2{e}^{-2t}$ is a solution of the differential equation $y^{″}\left(t\right)-k^{2}y\left(t\right)=0$ .

c.

To determine

To find: The general solution of the equation $y^{″}\left(t\right)-k^{2}y\left(t\right)=0$ for arbitrary $k>0$ and verify the conjecture.

d.

To determine

To verify: The given function $y\left(t\right)=C_1\cosh kt+C_2\sinh kt$ is a solution of the differential equation $y^{″}\left(t\right)-k^{2}y\left(t\right)=0$.