Change of variables in a Bernoulli equation The equation y'(t) + ay = by", where a, b, and p are real numbers, is called a Bernoulli equation. Unless p = 1, the equation is nonlinear and would appear to be difficult to solve-except for a small miracle. Through the change of variables v(t) = (y(t))'-P, the equation can be made linear. Carry out the following steps. a. Letting v = y!-P, show that y'(t) = y(1)P -v'(t). b. Substitute this expression for y' (t) into the differential equation and simplify to obtain the new (linear) equation v'(t) = a(1 – p)v = b(1 – p), which can be solved using the methods of this section. The solution y of the original equa- tion can then be found from v.
Change of variables in a Bernoulli equation The equation y'(t) + ay = by", where a, b, and p are real numbers, is called a Bernoulli equation. Unless p = 1, the equation is nonlinear and would appear to be difficult to solve-except for a small miracle. Through the change of variables v(t) = (y(t))'-P, the equation can be made linear. Carry out the following steps. a. Letting v = y!-P, show that y'(t) = y(1)P -v'(t). b. Substitute this expression for y' (t) into the differential equation and simplify to obtain the new (linear) equation v'(t) = a(1 – p)v = b(1 – p), which can be solved using the methods of this section. The solution y of the original equa- tion can then be found from v.
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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Transcribed Image Text:Change of variables in a Bernoulli equation The equation
y'(t) + ay = by", where a, b, and p are real numbers, is called a
Bernoulli equation. Unless p = 1, the equation is nonlinear and
would appear to be difficult to solve-except for a small miracle.
Through the change of variables v(t) = (y(t))'-P, the equation
can be made linear. Carry out the following steps.
a. Letting v = y!-P, show that y'(t) =
y(1)P
-v'(t).
b. Substitute this expression for y' (t) into the differential
equation and simplify to obtain the new (linear) equation
v'(t) = a(1 – p)v = b(1 – p), which can be solved using
the methods of this section. The solution y of the original equa-
tion can then be found from v.
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