Chapter 7.3, Problem 27E

a.

To determine

To draw : the direction field by using computer algebra system and sketch some solution curves by using direction field.

Explanation of Solution

Given information :

The differential equation is $y\text{'}=y^2$ .

Graph : using computer algebra system the direction field can be obtained as:

  

Start at the origin and move to the right in the direction of the line segment(which has slope $1$ ) then continue to draw the solution curve so that it moves parallel to the nearby line segments. For the members of this family of solution change the y -intercepts and draw.

The graph for members of solutions can be obtained as:

  

Interpretation: from the above graph it can be observed that when the y -intercept increases the graph curves increases rapidly.

b.

To determine

To solve : the differential equation.

Answer to Problem 27E

The solution of differential equation is $y=\frac{1}{c-x}$ .

Explanation of Solution

Given information :

The differential equation is $y\text{'}=y^2$ .

Calculation : here the equation is separable, so write the equation in the terms of differentials and then integrate both sides,

  $\begin{array}{l}y\text{'}=y^2\\ \frac{dy}{dx}=y^2\text{ [}y\text{'}=\frac{dy}{dx}]\\ \frac{dy}{y^2}=dx\text{ [separate the variables]}\\ {\displaystyle \int \frac{dy}{y^2}}={\displaystyle \int dx\text{ [integrate both side]}}\\ -\frac{1}{y}=x+c\text{ [simplify integrate] }\\ y=\frac{1}{c-x}\text{ [simplify]}\end{array}$

Here $c$ is a arbitrary constant.

Hence, the solution of differential equation is $y=\frac{1}{c-x}$ .

c.

To determine

To graph : the serval members of the family of solution obtained in part (b).

Explanation of Solution

Given information :

The differential equation is $y\text{'}=y^2$ .

graph : from pat (b) it can be observed that the solution of differential equation is $y=\frac{1}{c-x}$ .

Now substitute different value for c to get different members of family of solutions.

  $\begin{array}{l}y=\frac{1}{1-x}\text{ [}c=1]\\ y=\frac{1}{2-x}\text{ [}c=2]\\ y=\frac{1}{3-x}\text{ [}c=3]\\ y=\frac{1}{-2-x}\text{ [}c=-2]\\ y=\frac{1}{-1-x}\text{ [}c=-1]\end{array}$

Using implicit plotting capability of a CAS to graph the curve $y=\frac{1}{1-x},y=\frac{1}{2-x},y=\frac{1}{3-x},y=\frac{1}{-2-x},y=\frac{1}{-1-x}$ .

The graph is obtained as:

  

Interpretation:

from the above graph and part (a) it can be observed that the curves of the members of the family of solution of differential equation is same.