A Bernoulli differential equation is oneof the formdy+ P(x)y = Q(x)y".dxObserve that, if n0 or 1, the Bernoulliequation is linear. For other values of n, thesubstitution u = y'-n transforms the Bernoulliequation into the linear equationdu+ (1 — п)Р(х)и %3D(1 — п)Q(»).dxUse an appropriate substitution to solve theequation8.and find the solution that satisfies y(1) = 1.y(x) =

Given - A Bernoulli Differential equation is one of the form dydx + Pxy = Qxyn . Observe that , if n…

Answered: A Bernoulli differential equation is…

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
+ P(x)y = Q(x)y".
dx
Observe that, if n
0 or 1, the Bernoulli
equation is linear. For other values of n, the
substitution u = y'-n transforms the Bernoulli
equation into the linear equation
du
+ (1 — п)Р(х)и %3D(1 — п)Q(»).
dx
Use an appropriate substitution to solve the
equation
8.
and find the solution that satisfies y(1) = 1.
y(x) =

Expert Solution

Step 1

Given - A Bernoulli Differential equation is one of the form $\frac{d y}{d x} + P (x) y = Q (x) y^{n}$ . Observe that , if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution $u = y^{1 - n}$ transforms the Bernoulli equation into the linear equation $\frac{d u}{d x} + (1 - n) P (x) u = (1 - n) Q (x)$

To find - Use an appropriate substitution to solve the equation $y' - \frac{8}{x} y = \frac{y^{4}}{x^{2}}$ , and find the solution that satisfies $y (1) = 1$

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