Consider the initial value problem Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution uyl-n transforms the Bernoulli equation into the linear equation (a) This differential equation can be written in the form (*) with P(x) = Q(x) = n = (b) The substitution u = du dx + , and u= (e) Finally, solve for y. y(x) = dy dx du dx + P(x)y = Q(x)y" (*) + (1 − n)P(x)u = (1 − n)Q(x). xy + y = 4xy², y(1) = 9. 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) =
Consider the initial value problem Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution uyl-n transforms the Bernoulli equation into the linear equation (a) This differential equation can be written in the form (*) with P(x) = Q(x) = n = (b) The substitution u = du dx + , and u= (e) Finally, solve for y. y(x) = dy dx du dx + P(x)y = Q(x)y" (*) + (1 − n)P(x)u = (1 − n)Q(x). xy + y = 4xy², y(1) = 9. 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) =
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
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Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:Consider the initial value problem
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
(a) This differential equation can be written in the form (*) with
P(x) =
Q(x) =
n =
(b) The substitution u =
du
dx
+
and
u=
(e) Finally, solve for y.
y(x) =
dy
dx
du
dx
+ P(x)y = Q(x)y" (*)
+ (1 − n)P(x)u = (1 − n)Q(x).
xy + y = 4xy², y(1) = 9.
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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