Chapter 6.1, Problem 48E

(a)

To determine

To sketch: The graph of the solution to the given initial value problem.

Answer to Problem 48E

The graph is:

  

Explanation of Solution

Given information:

The differential equation is: $\frac{dy}{dx}=-x^{-2}$

And $y=1$ when

  $x=1$

Calculation:

by resolving its anti derivative, one must locate the differential equation's general solution. At this time,

  $\begin{array}{c}\frac{dy}{dx}=-x^{-2}\\ \Rightarrow y=\int \left(-x^{-2}\right)dx\\ \Rightarrow y=x^{-1}+C\end{array}$

Put the initial condition $y=1$ when

  $x=1$ to find $C$ :

  $\begin{array}{c}1=1^{-1}+C\\ \Rightarrow 1=1+C\\ \Rightarrow 0=C\end{array}$

To determine the specific solution of the differential equation, substitute the value of $C$ into the general solution.

  $\begin{array}{c}y=x^{-1}+0\\ \Rightarrow y=x^{-1}\end{array}$

When a curve is discontinuous, the initial condition only provides with the portion of the curve that is continuous and traverses the given point. Consequently, only graph the right half of the piece that passes through when $y=x^{-1}$ is plotted $(1,1)$ .

  

(b)

To determine

To explain The given graph is not the correct answer to part $(a)$ .

Answer to Problem 48E

Because the graph also includes the section of the graph to the left of $x=0$ , it is then inaccurate.

Explanation of Solution

Given information:

The graph is:

  

Explain:

The initial condition only provides  with the portion of the graph that traverses the point when a curve is discontinuous. At $x=0$ , the curve has a discontinuity. Therefore, since $(1,1)$ is located to the right of $x=0$, it only gives the part of the curve that is above $x=0$ . Because the graph also includes the section of the graph to the left of $x=0$ , it is then inaccurate.