=y-2y³, y(0) = 1. dt get to translate back to the desi
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![16. We have seen that Bernoulli's change of variable \( z(t) = 1/y(t) \) transforms the nonlinear logistic equation
\[
\frac{dy}{dt} = ay - by^2
\]
into a linear equation. There is a more general version of this change of variables. Consider the equation
\[
\frac{dy}{dt} = ay - by^n,
\]
where \( n \) is any (positive or negative) integer, and \( a, b \) are constants. Let \( z(t) = y(t)^{1-n} \), and find the equation satisfied by \( z \). This equation should be linear. Use this change of variables to solve the initial value problem
\[
\frac{dy}{dt} = y - 2y^3, \quad y(0) = 1.
\]
(After you find \( z(t) \), don't forget to translate back to the desired solution \( y(t) \).)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61f2bd8c-527a-479e-9deb-09c663979694%2F2097a35b-35cc-4568-aea8-80b300fa074b%2F0f8qkpi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:16. We have seen that Bernoulli's change of variable \( z(t) = 1/y(t) \) transforms the nonlinear logistic equation
\[
\frac{dy}{dt} = ay - by^2
\]
into a linear equation. There is a more general version of this change of variables. Consider the equation
\[
\frac{dy}{dt} = ay - by^n,
\]
where \( n \) is any (positive or negative) integer, and \( a, b \) are constants. Let \( z(t) = y(t)^{1-n} \), and find the equation satisfied by \( z \). This equation should be linear. Use this change of variables to solve the initial value problem
\[
\frac{dy}{dt} = y - 2y^3, \quad y(0) = 1.
\]
(After you find \( z(t) \), don't forget to translate back to the desired solution \( y(t) \).)
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