Chapter 7.4, Problem 6E

To determine

To solve initial value problem using variable separation.

Answer to Problem 6E

The solution of differential equation $\frac{dy}{dx}={\cos }^{2}y$ is $y={\tan }^{-1}x$ and the domain over which the solution is valid is $\left(-\infty ,\infty \right)$ that is for all real values of $x$ .

Explanation of Solution

Given information:

The differential equation is $\frac{dy}{dx}={\cos }^{2}y$ and the initial condition is $y=0\text{ when }x=0$ .

Calculation:

The differential equation is $\frac{dy}{dx}={\cos }^{2}y$

Using separation of variables method

  $\begin{array}{l}\frac{dy}{dx}={\cos }^{2}y\\ \frac{1}{{\cos }^{2}y}dy=dx\\ {\sec }^{2}ydy=dx\\ \text{integrating both side}\\ {\displaystyle \int {\sec }^{2}ydy}={\displaystyle \int dx}\\ \tan y=x\\ y={\tan }^{-1}x+C\\ \text{initial condition }y=0\text{ when }x=0\\ 0={\tan }^{-1}0+C\\ C=0\\ y={\tan }^{-1}x\\ \text{Domain: }\left(-\infty ,\infty \right)\end{array}$

Therefore, the solution of differential equation $\frac{dy}{dx}={\cos }^{2}y$ is $y={\tan }^{-1}x$ and the domain over which the solution is valid is $\left(-\infty ,\infty \right)$ that is for all real values of $x$ .