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A Bernoulli differential equation is one of the form dy + P(x)y= Q(x)y" (*) da y ¹ Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = into the linear equation du + (1 − n)P(x)u = (1-n)Q(x). da Consider the initial value problem xy + y = 8xy², y(1) = 6. (a) This differential equation can be written in the form (*) with P(x) Q(x) and n transforms the Bern

A Bernoulli differential equation is one of the form dy + P(x)y= Q(x)y" (*) da y ¹ Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = into the linear equation du + (1 − n)P(x)u = (1-n)Q(x). da Consider the initial value problem xy + y = 8xy², y(1) = 6. (a) This differential equation can be written in the form (*) with P(x) Q(x) and n transforms the Bern

Advanced Engineering Mathematics
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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A Bernoulli differential equation is one of the form dy da + P(x)y= Q(x)y" (*) Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = y¹¹ transforms the Bernoulli equation into the linear equation du + (1 − n)P(x)u = (1 − n)Q(x). dr Consider the initial value problem xy + y = 8ry², y(1) = 6. (a) This differential equation can be written in the form (*) with P(x) = 1 Q(x) and
Transcribed Image Text:A Bernoulli differential equation is one of the form dy da + P(x)y= Q(x)y" (*) Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = y¹¹ transforms the Bernoulli equation into the linear equation du + (1 − n)P(x)u = (1 − n)Q(x). dr Consider the initial value problem xy + y = 8ry², y(1) = 6. (a) This differential equation can be written in the form (*) with P(x) = 1 Q(x) and
(b) The substitution u = will transform it into the linear equation du + da (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) = m (e) Finally, solve for y. y(x) =
Transcribed Image Text:(b) The substitution u = will transform it into the linear equation du + da (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) = m (e) Finally, solve for y. y(x) =
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