A Bernoulli differential equation is one of the form dy dî Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = y¹-n transforms the Bernoulli equation into the linear equation Consider the initial value problem n = (b) The substitution u = du dx xy' + y = −8xy², y(1)=—7. (a) This differential equation can be written in the form (*) with P(x) = Q(x) = + du dx and + P(x)y = Q(x)y” (*) U= (e) Finally, solve for y. y(x) = + (1 − n)P(x)u = (1 − n)Q(x). will transform it into the linear equation (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). u(x) =
A Bernoulli differential equation is one of the form dy dî Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = y¹-n transforms the Bernoulli equation into the linear equation Consider the initial value problem n = (b) The substitution u = du dx xy' + y = −8xy², y(1)=—7. (a) This differential equation can be written in the form (*) with P(x) = Q(x) = + du dx and + P(x)y = Q(x)y” (*) U= (e) Finally, solve for y. y(x) = + (1 − n)P(x)u = (1 − n)Q(x). will transform it into the linear equation (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). u(x) =
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
dx
Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution
u = y¹n transforms the Bernoulli equation into the linear equation
1-n
Consider the initial value problem
n =
(b) The substitution u =
du
dx
xy + y = −8xy², y(1) = −7.
(a) This differential equation can be written in the form (*) with
P(x) =
=
Q(x) =
+
du
dx
and
+ P(x)y= Q(x)y" (*)
U =
(e) Finally, solve for y.
y(x) =
+ (1 − n)P(x)u = (1 − n)Q(x).
will transform it into the linear equation
(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).
u(x) =
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