Chapter 7.3, Problem 35E

To determine

To solve : the integral equation.

Answer to Problem 35E

the solution of integral equation is $y={\left(\frac{x}{2}+2\right)}^2$ .

Explanation of Solution

Given information :

The integral equation is $y\left(x\right)=4+{\displaystyle {\int }_0^{x}2t\sqrt{y\left(t\right)}}dt$ .

Calculation: to solve the integral equation first find the initial condition of the equation,

For initial equation substitute $x=0,t=0$ in the integral equation,

  $\begin{array}{l}y\left(x\right)=4+{\displaystyle {\int }_0^{x}2t\sqrt{y\left(t\right)}}dt\\ y\left(0\right)=4+{\displaystyle {\int }_0^{x}2·0·\sqrt{y\left(0\right)}}dt\text{ [substitute]}\\ y\left(0\right)=4\text{[simplify]}\end{array}$

Now solve the integral equation, differentiate the equation then it will be converted into differential equation then solve the differential equation,

  $\begin{array}{l}y\left(x\right)=4+{\displaystyle {\int }_0^{x}2t\sqrt{y\left(t\right)}}dt\\ y\text{'}\left(x\right)=0+2x\sqrt{y\left(x\right)}\text{ [differentiate with respect to }x\text{ and put }t=x]\\ y\text{'}\left(x\right)=2x\sqrt{y\left(x\right)}\end{array}$

So the differential equation for integral equation is $y\text{'}\left(x\right)=2x\sqrt{y\left(x\right)}$ .

Now solve the differential equation,

Let $y\left(x\right)=y$

  $\begin{array}{l}y\text{'}\left(x\right)=2x\sqrt{y\left(x\right)}\\ y\text{'}=2x\sqrt{y}\text{ [subtitute }y\left(x\right)=y]\\ \frac{dy}{dx}=2x\sqrt{y}\text{ [}y\text{'}=\frac{dy}{dx}]\\ \frac{dy}{\sqrt{y}}=2xdx\text{ [separate variables]}\\ {\displaystyle \int \frac{dy}{\sqrt{y}}={\displaystyle \int 2xdx}}\text{ [integrate both side]}\\ 2\sqrt{y}=x+c\text{ [simplify integrate]}\\ \sqrt{y}=\frac{x}{2}+c\text{ [multiply by 2]}\\ y={\left(\frac{x}{2}+c\right)}^2\text{ [simplify}]\end{array}$

Where c are arbitrary constant.

Since $y\left(0\right)=4$ , so

  $\begin{array}{l}y={\left(\frac{x}{2}+c\right)}^2\\ 4={\left(\frac{0}{2}+c\right)}^2\text{ [substitue]}\\ 4=c^2\text{ [sqaure]}\\ c=2\text{ [sqaure root]}\end{array}$

Thus the solution of integral equation is

  $y={\left(\frac{x}{2}+2\right)}^2$