Chapter 7.1, Problem 23E

To determine

To solve the initial value problem using fundamental theorem.

Answer to Problem 23E

The solution of differential equation is $y={\displaystyle \underset{2}{\overset{x}{\int }}{e}^{\cos t}}dt+\text{ 9}$ .

Explanation of Solution

Given information:

It is given that

  $F^{\prime }\left(x\right)=e^{\cos x}\text{ and }F\left(2\right)=9$

Calculation:

According to fundamental theorem ${\displaystyle \underset{a}{\overset{x}{\int }}{F}^{\prime }\left(x\right)}dt=\text{ F}\left(x\right)-F\left(a\right)$ .This can also be written as $F\left(x\right)={\displaystyle \underset{a}{\overset{x}{\int }}{F}^{\prime }\left(t\right)}dt+\text{ F}\left(a\right)$ . The initial condition given is $F\left(2\right)=9$ .

Let $a=2$ , then $F\left(a\right)=F\left(2\right)=9$ , also it is given that $F^{\prime }\left(x\right)=\frac{dy}{dx}=e^{\cos x}$ .

Substituting $F\left(x\right)=y$ , $F\left(a\right)=F\left(2\right)=9$ and $F^{\prime }\left(x\right)=e^{\cos x}$

therefore

  $\begin{array}{l}F\left(x\right)={\displaystyle \underset{a}{\overset{x}{\int }}{F}^{\prime }\left(t\right)}dt+\text{ F}\left(a\right)\\ y={\displaystyle \underset{2}{\overset{x}{\int }}{e}^{\cos t}}dt+\text{ 9}\end{array}$