Chapter 6.1, Problem 59E

(a)

To determine

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

Answer to Problem 59E

The correct graph is $(b)$

  

Explanation of Solution

Given information:

The differential equation is:

  $\begin{array}{c}\frac{dy}{dx}=\frac{1}{1+x^2},\\ y(0)=\frac{\pi }{2}\end{array}$

Calculation:

Graph $(a)$ has a slope of 0 at $x=0$

Graph $(b)$ has a positive slope has $x=0$

Graph $(c)$ has a negative slope at $x=0$

Therefore, put $x=0$ into the differential equation

  $\frac{dy}{dx}=\frac{1}{1+x^2}$

To find which graph has the correct slope:

  $\begin{array}{c}\frac{dy}{dx}=\frac{1}{1+0^2}\\ =1\end{array}$

The required correct graph is then $(b)$ since its the only one with a positive slope at $x=0$ .

(b)

To determine

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

Answer to Problem 59E

The slope is $+1$ , that means the answer cannot be neither $(a)$ nor $(c)$

Explanation of Solution

Given information:

The differential equation is:

  $\begin{array}{c}\frac{dy}{dx}=\frac{1}{1+x^2},\\ y(0)=\frac{\pi }{2}\end{array}$

Explain:

First find the slope of the graph at $x=0$ , using the derivative. The slope is $+1$ , that means the answer cannot be neither $(a)$ nor $(c)$