Chapter 7.2, Problem 2E

a.

To determine

To graph : the solutions that satisfy the initial conditions.

Explanation of Solution

Given information :

the differential equation is $y\text{'}=\tan (\frac{1}{2}\pi y)$ .

Graph :

From the direction field provided in the question it can be observed that,

The solution of differential equation is $\ln \left|\sin \pi y\right|=\frac{\pi x}{2}+c$ .

For initial conditions change the y -intercept,

Change the y -intercept $1,0.2,2,3$ .

To draw a graph, use plotting approach.

x	0	0	0	1
y	1	0.2	2	3

The graph is obtained as:

  

Interpretation : from the above graph it can be observed that the graph for all initial conditions y -intercept will change.

b.

To determine

To find : the all equilibrium solutions.

Answer to Problem 2E

The solution is $\ln \left|\sin \pi y\right|=\frac{\pi x}{2}$

Explanation of Solution

Given information :

the differential equation is $y\text{'}=\tan (\frac{1}{2}\pi y)$ .

Calculation : from the above part, and direction field

It can be observed that the equilibrium solutions obtained when the y -intercept is integer.

Thus, the solutions is $\ln \left|\sin \pi y\right|=\frac{\pi x}{2}$