   Chapter 8.3, Problem 41E

Chapter
Section
Textbook Problem

Slope Field In Exercises 41 and 42, use a computer algebra system to graph the slope field for the differential equation and graph the solution satisfying the specified initial condition. d y d x = 3 sin x y , y ( 0 ) = 2

To determine

To graph: The slope field for the provided differential equation and the solution satisfying the specified initial condition.

Explanation

Given: The provided differential equation is as follows;

dydx=3sinxy, y(0)=2

Graph:

Consider the provided differential equation dydx=3sinxy

Now rearrange this differential equation as follows;

dydx=3sinxy to get ydy=3sinxdx ---------(1)

Thus this differential equation has changed into variable separable form.

Now, integrate both sides of equation (1), we get

ydy=3sinxdx.

y22=3cosx+C -----------(2)

Now, use the provided initial conditions y(0)=2, we get y=2 when x=0,

Therefore,

222=3cos(0)+C.

C=5 -------------(3)

Now substitute the value of C from equation (3) to equation (2), we get

y2=106cosx, which is the required solution

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