Chapter 7.2, Problem 69E

To determine

To show that $y=\ln \left|\frac{\cos 3}{\cos x}\right|+5$ is the solution of the initial value problem $\frac{dy}{dx}=\tan (x),y(3)=5$

Explanation of Solution

Given information: $\frac{dy}{dx}=\tan (x),y(3)=5$

Formula used: ${\displaystyle \int \tan xdx=-\ln (\cos x)+C}$

Proof :

  $dy=\tan (x)dx$

Integrating with respect to x , the expression becomes as:

  $y=-\ln (\cos (x))+C$

Using, $y(3)=5$

  $5=-\ln (\cos (3))+C$

  $\Rightarrow C=\ln (\cos (3))+5$

The expression is $y=-\ln (\cos (x))+\ln (\cos (3))+5$

  $y=\ln \left(\frac{\cos (3)}{\cos (x)}\right)+5$

Hence, $y=\ln \left|\frac{\cos 3}{\cos x}\right|+5$ is the solution of $\frac{dy}{dx}=\tan (x)$