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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Chapter
Section
BuyFindarrow_forward

Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
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

To determine

To draw:

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

Explanation

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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