Chapter 6.2, Problem 26E

To determine

(a)

To show:

The value of $C_1$ is $0$ for $C_1{e}^{r_{1}x}+C_2{e}^{r_{2}x}+\mathrm{...}+C_n{e}^{r_{n}x}=0$, $r_1>r_2>\mathrm{...}>r_n$ when $x\to +\infty$.

To determine

(b)

To show:

The value of $C_2$ is $0$ for $C_2{e}^{r_{2}x}+C_3{e}^{r_{3}x}\mathrm{...}+C_n{e}^{r_{n}x}=0$, $r_1>r_2>\mathrm{...}>r_n$ when $x\to +\infty$.

To determine

(c)

To prove:

All the coefficients, $C_1,C_2,\mathrm{...},C_n$ are zero and hence $e^{r_{1}x},e^{r_{2}x},\mathrm{........},e^{r_{n}x}$ are linearly independent on $\left(-\infty ,\infty \right)$.