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Numerade Educator

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Problem 28 Easy Difficulty

(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' = xy $

Answer

a) SEE GRAPH
b) $y=A c^{x^{2} / 2}$
c) SEE GRAPH

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Video Transcript

for this problem, we are asked to use a computer algebra system to draw direction field for the differential equation why prime of X equals X times Y. Then to use to sketch some solution curves without solving the differential equation for B. Were also find the solution, and then for C, were asked to use the CS to draw several members of the family of solutions. So here I can just get a Wolfram mathematica to solve this directly. So plotted down here, we have in a second it will be the direction field once it re computes. All right, so here we can see the direction field, we can see that we stay at zero, long zero, but we have this sort of almost parabolic type shape. Um in addition for part B we have through what I'm using here the explicit solution where I can even show you these step by step. So it is a separable equation where we end up having a Y prime over Y equals x integral of Y prime over why is log of X and integral of X is x squared over two plus C. One giving us the solution that we arrive at. Then lastly we have shown down here is an example of a sample solution family