dy16. We have seen that Bernoulli's change of variable z(t) = 1/y(t) transforms the nonlinearlogistic equation ay-by² into a linear equation. There is a more general version ofthis change of variables. Consider the equationdtdydt=ay - by",where n is any (positive or negative) integer, and a, b are constants. Let z(t) = y(t)¹-",and find the equation satisfied by z. This equation should be linear. Use this change ofvariables to solve the initial value problemdy =y - 2y³,dt(After you find z(t), don't forget to translate back to the desired solution y(t).)y(0)== 1.

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=y-2y³, y(0) = 1. dt get to translate back to the desi

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.6: The Inverse Trigonometric Functions

Problem 94E

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Question

dy
16. We have seen that Bernoulli's change of variable z(t) = 1/y(t) transforms the nonlinear
logistic equation ay-by² into a linear equation. There is a more general version of
this change of variables. Consider the equation
dt
dy
dt
=ay - by",
where n is any (positive or negative) integer, and a, b are constants. Let z(t) = y(t)¹-",
and find the equation satisfied by z. This equation should be linear. Use this change of
variables to solve the initial value problem
dy =y - 2y³,
dt
(After you find z(t), don't forget to translate back to the desired solution y(t).)
y(0)=
= 1.

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