 # 27–28 (a) Use a computer algebra system to draw a direction field for the differential equation. Get a printout and use it to sketch some solution curves without solving the differential equation. (b) Solve the differential equation. (c) Use the CAS to draw several members of the family of solutions obtained in part (b). Compare with the curves from part (a). y ′ = x y ### Calculus (MindTap Course List)

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285740621 ### Calculus (MindTap Course List)

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285740621

#### Solutions

Chapter
Section
Chapter 9.3, Problem 28E
Textbook Problem

## 27–28(a) Use a computer algebra system to draw a direction field for the differential equation. Get a printout and use it to sketch some solution curves without solving the differential equation.(b) Solve the differential equation.(c) Use the CAS to draw several members of the family of solutions obtained in part (b). Compare with the curves from part (a). y ′ = x y

Expert Solution
To determine

To draw:

A direction field for the given differential equation and then sketch some solution curves without solving the differential equation.

### Explanation of Solution

1) Concept:

Use Mathematica to draw a direction field and then sketch some solution curves with your hand.

2) Given:

y'=xy

3) Calculation:

To draw direction field, use below command in Mathematica

dfield=VectorPlot[{1,x*y},{x,-3,3},{y,-3,3},AxesTrue,AxesLabel{"x","dydx=x*y"},VectorPointsFine,VectorStyle"Segment"]

then output is

Now, draw some solution curves with hand without solving the differential equation is

Conclusion:

The graph of directional field using Mathematica is

And graph of some solution curves with hand without solving the differential equation is

To solve:

The given differential equation y'=xy

Solution:

y=Kex22 & y=0

1) Concept:

To solve the differential equation of the form dydx=gxhy , rewrite the equation in the differential form  hydy=gxdx so that all  y’s are on one side and all  x’s are on the other side

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