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(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 $

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

Calculus 2 / BC

Chapter 9

Differential Equations

Section 3

Separable Equations

Campbell University

Oregon State University

Harvey Mudd College

Lectures

13:37

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.

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(a) Use a computer algebra…

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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

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