Chapter 7.7, Problem 15P

(a)

To determine

To show: That the solution $\text{x}=\varphi \left(t\right)$ exists and satisfy the integral equation $\varphi \left(t\right){\text{=x}}^0+{\displaystyle {\int }_0^t\text{A}\varphi \left(s\right)ds}$.

(b)

To determine

To show: That the equation ${\varphi }^{\left(1\right)}\left(t\right)=\left(\text{I}+\text{A}t\right){\text{x}}^0$.

(c)

To determine

To show: That the inductive argument ${\varphi }^n\left(t\right)=\left(\text{I}+\text{A}t+\text{A}\frac{t^2}{2!}+\cdot \cdot \cdot +{\text{A}}^n\frac{t^n}{n!}\right){\text{x}}^0$.

(d)

To determine

To show: That the solution of the initial value problem $\text{x}\prime =\text{Ax},\text{ x}\left(0\right)={\text{x}}^0$ is $\varphi \left(t\right)=\exp \left(\text{A}t\right){\text{x}}^0$.