Chapter 7.1, Problem 27E

To determine

To match the differential equation with the graph of a family of functions

Answer to Problem 27E

Option $\left(a\right)$ represents the graph of the differential equation $\frac{dy}{dx}={\left(\cos x\right)}^2$ .

Explanation of Solution

Given information:

The differential equation given is $\frac{dy}{dx}={\left(\cos x\right)}^2$

The graphs given are

  

Calculation:

The differential equation given is $\frac{dy}{dx}={\left(\cos x\right)}^2$ . The slope at any point $\left(0,y\right)$ is ${\left(\cos 0\right)}^2=1$ therefore along y-axis there are tiny segments with slope $1$ .The slope at any points $\left(\pi ,y\right)$ or $\left(-\pi ,y\right)$ will be $-1$ therefore there are tiny segments along vertical lines $x=\pi$ and $x=-\pi$ .

The slope at odd multiples of $\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ will be zero therefore there are tiny horizontal lines at odd multiples of $\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ . There are tiny segments of slope $1$ along the line $x=\pm 2\pi$ . As the slope ${\left(\cos x\right)}^2$ contains even power therefore the graph will not be much flatten it will be more slanted.

Hence option $\left(a\right)$ represents the graph of the differential equation $\frac{dy}{dx}={\left(\cos x\right)}^2$ .