Chapter 6.1, Problem 60E

(a)

To determine

To choose: The correct graph that shows the solution of the initial value without actually solving problem.

Answer to Problem 60E

The correct graph is $(b)$

  

Explanation of Solution

Given information:

The differential equation is:

  $\begin{array}{c}\frac{dy}{dx}=-x,\\ y(-1)=1\end{array}$

Calculation:

For $\frac{dy}{dx}=-x$

  $\frac{dy}{dx}<0$ when $x>0$

And $\frac{dy}{dx}>0$ when $x<0$

Only graph $(b)$ satisfies this , hence the graph $(a)$ has $\frac{dy}{dx}<0$ for all $x$ .

And the graph $(c)$ has $\frac{dy}{dx}>0$ for $x>0$

And $\frac{dy}{dx}<0$ for $x<0$

Also find the correct graph by just looking at the slope at $x=-1$ .

When $x=-1$ ,

  $\begin{array}{c}\frac{dy}{dx}=-(-1)\\ =1\end{array}$

So the slope must be positive. Graphs $(a)$ and $(c)$ have negatives slopes at $x=-1$

So they are incorrect leaving graph $(b)$ as the correct answer.

(b)

To determine

To explain: The reason that how eliminated two of the possibilities

Answer to Problem 60E

the slope must be positive. Graphs $(a)$ and $(c)$ have negatives slopes at $x=-1$ so they can be eliminated.

Explanation of Solution

Given information:

The differential equation is:

  $\begin{array}{c}\frac{dy}{dx}=-x,\\ y(-1)=1\end{array}$

Explain:

For $\frac{dy}{dx}=-x$

  $\frac{dy}{dx}<0$ when $x>0$

And $\frac{dy}{dx}>0$ when $x<0$

Only graph $(b)$ satisfies this , hence the graph $(a)$ has $\frac{dy}{dx}<0$ for all $x$ .

And the graph $(c)$ has $\frac{dy}{dx}>0$ for $x>0$

And $\frac{dy}{dx}<0$ for $x<0$ so they can be eliminated.

Also find the correct graph by just looking at the slope at $x=-1$ .

When $x=-1$ ,

  $\begin{array}{c}\frac{dy}{dx}=-(-1)\\ =1\end{array}$

So the slope must be positive. Graphs $(a)$ and $(c)$ have negatives slopes at $x=-1$

So they can be eliminated.