A Bernoulli differential equation is one of the form dy + P(х)у %3D Q(х)у" (*) dx bserve that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = y-" transforms the Bernoulli equation into the linear equation du + (1 — п)Р(х)и %3D (1 — п)Q(х). dx onsider the initial value problem xy + y = -6xy², y(1) = -1. This differential equation can be written in the form (*) with (х) %3D (x) = and The substitution u = will transform it into the linear equation u u = Using the substitution in part (b), we rewrite the initial condition in terms of x and u: 1) = O Now solve the linear equation in part (b), and find the solution that satisfies the initial condition in part (c). x) = Finally, solve for y.
A Bernoulli differential equation is one of the form dy + P(х)у %3D Q(х)у" (*) dx bserve that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = y-" transforms the Bernoulli equation into the linear equation du + (1 — п)Р(х)и %3D (1 — п)Q(х). dx onsider the initial value problem xy + y = -6xy², y(1) = -1. This differential equation can be written in the form (*) with (х) %3D (x) = and The substitution u = will transform it into the linear equation u u = Using the substitution in part (b), we rewrite the initial condition in terms of x and u: 1) = O Now solve the linear equation in part (b), and find the solution that satisfies the initial condition in part (c). x) = Finally, solve for y.
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:A Bernoulli differential equation is one of the form
dy
+ P(х)у %3D 0(х)у" (*)
dx
Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = y'¬" transforms the Bernoulli equation into the linear equation
du
+ (1 — п)Р(х)и — (1 — п)Q(х).
dx
Consider the initial value problem
ху + у%3D -6ху", у(1) — —1.
(a) This differential equation can be written in the form (*) with
P(x) =
Q(x) =
, and
n =
(b) The substitution u =
will transform it into the linear equation
du
u =
dx
(c) Using the substitution in part (b), we rewrite the initial condition in terms of x and u:
u(1) =
(d) Now solve the linear equation in part (b), and find the solution that satisfies the initial condition in part (c).
= (x)n
(e) Finally, solve for y.
У(x) —
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