Chapter 7.1, Problem 7QR

To determine

To determine whether or not the function y satisfies the differential equation.

Answer to Problem 7QR

  $\text{ y}\tan x=\sec x\tan x,\text{ the function satisfy the differential equation}\text{.}$

Explanation of Solution

Given information: $\text{The given function is, y}=\sec x.$

Concept used: Differentiation.

  $\begin{array}{l}\text{The process of finding the derivative}\text{.}\\ \frac{\text{dy}}{\text{d}x}=\underset{\text{δ}x\to \text{0}}{\text{lim}}\frac{\text{δy}}{\text{δ}x}=\underset{\text{δ}x\to \text{0}}{\text{lim}}\frac{f\left(x+\text{δ}x\right)-f\left(x\right)}{\text{δ}x}\end{array}$

Calculation: The function y satisfies the differential equation as,

  $\begin{array}{l}\frac{\text{dy}}{\text{d}x}=\text{y}\tan x\text{ , y}=\sec x\\ \text{Substitute the value of y into the differential equation}\text{.}\\ \frac{\text{d}}{\text{d}x}\left(\text{y}\right)=\frac{\text{d}}{\text{d}x}\left(\sec x\right)\\ =\sec x.\tan x\\ =\text{y}\tan x\left(\text{y}=\sec x\right)\\ \text{Since, y}\tan x=\sec x\tan x,\text{ the function satisfy the differential equation}\text{.}\end{array}$