Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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 \( y(t) \).)
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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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