Chapter 7.3, Problem 28E

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 $y\text{'}=xy$ .

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 shrink.

b.

To determine

To solve : the differential equation.

Answer to Problem 28E

The solution of differential equation is $y=A{e}^{\frac{x^2}{2}}$ .

Explanation of Solution

Given information :

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

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

  $\begin{array}{l}y\text{'}=xy\\ \frac{dy}{dx}=xy\text{ [}y\text{'}=\frac{dy}{dx}]\\ \frac{dy}{y}=xdx\text{ [separate the variables]}\\ {\displaystyle \int \frac{dy}{y}}={\displaystyle \int xdx\text{ [integrate both side]}}\\ \ln \left|y\right|=\frac{x^2}{2}+c\text{ [simplify integrate] }\\ y=A{e}^{\frac{x^2}{2}}\text{ [take ln both side and }{e}^c\text{=}A\text{]}\end{array}$

Here A , $c$ is a arbitrary constant.

Hence, the solution of differential equation is $y=A{e}^{\frac{x^2}{2}}$ .

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 $y=A{e}^{\frac{x^2}{2}}$ .

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

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

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

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.