Chapter 7.2, Problem 1E

a.

To determine

To graph : the solutions that satisfy the initial conditions.

Explanation of Solution

Given information :

the differential equation is $y\text{'}=x\cos \pi y$ .

Graph :

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

The solution of differential equation is $\ln |sex\pi y+\tan \pi y|=\frac{\pi x^2}{2}+c$ .

So now for the initial conditions, change the y -intercept of the solution of the differential equation,

Substitute c by $0,1,0.5,1.6$ .

All the solutions are equilibrium solutions, so use plotting approach to make graph,

x	0	0	0	0
y	0	1	0.5	1.6

The graph is obtained as:

  

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

b.

To determine

To find : the all equilibrium solutions.

Answer to Problem 1E

The solution is $\ln |sex\pi y·\tan \pi y|=\frac{\pi x^2}{2}$

Explanation of Solution

Given information :

the differential equation is $y\text{'}=x\cos \pi y$ .

Calculation : from the above part ,

It can be observed that for the given initial conditions the differential equation has equilibrium solutions.

Thus, the solutions is $\ln |sex\pi y·\tan \pi y|=\frac{\pi x^2}{2}$