Chapter 7.1, Problem 8QR

To determine

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

Answer to Problem 8QR

  ${\text{ y}}^2=\frac{1}{x^2}\ne -\frac{1}{x^2},\text{ the function does not satisfy the differential equation}\text{.}$

Explanation of Solution

Given information: $\text{The given function is, y}=x^{-1}.$

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,

  $\begin{array}{l}\frac{\text{dy}}{\text{d}x}={\text{y}}^2\text{ , y}=x^{-1}\\ \text{Substitute the value of y into the differential equation}\text{.}\\ \frac{\text{d}}{\text{d}x}\left(\text{y}\right)=\frac{\text{d}}{\text{d}x}\left(x^{-1}\right)\\ =\left(-1\right)x^{-1-1}\\ =-x^{-2}\\ =-\frac{1}{x^2}\\ {\text{y}}^2={\left(x^{-1}\right)}^2\\ =x^{-2}\\ =\frac{1}{x^2}\\ {\text{Since, y}}^2=\frac{1}{x^2}\ne -\frac{1}{x^2},\text{ the function does not satisfy the differential equation}\text{.}\end{array}$