(a) Prove that the substitution ? = ?1−? reduces the Bernoulli’s equation????+ ?(?)? = ?(?)??, ? ≠ 0 , ? ≠ 1to a linear equation in ?.(b) Find the solution of the nonlinear IVP?2 ????− 2?? = 3?3, ?(1) = 1. 2. (a) Prove that the substitution u = yl¬n reduces the Bemoulli's equation1-n+ P(x)y = f(x)y", n ± 0 , n ± 1dxto a linear equation in u.(b) Find the solution of the nonlinear IVPdyx2- 2xydx3y³,y(1) = 1.

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2. (a) Prove that the substitution u = yl¬n reduces the Bemoulli's equation + P(x)y = f(x)y", n ± 0 , n # 1 dy dx to a linear equation in u. (b) Find the solution of the nonlinear IVP dy 2xy = 3y³, y(1) = 1. dx

2. (a) Prove that the substitution u = yl¬n reduces the Bemoulli's equation + P(x)y = f(x)y", n ± 0 , n # 1 dy dx to a linear equation in u. (b) Find the solution of the nonlinear IVP dy 2xy = 3y³, y(1) = 1. dx

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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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

(a) Prove that the substitution ? = ?1−? reduces the Bernoulli’s equation
??
??
+ ?(?)? = ?(?)??, ? ≠ 0 , ? ≠ 1
to a linear equation in ?.
(b) Find the solution of the nonlinear IVP
?2 ??
??
− 2?? = 3?3, ?(1) = 1.

2. (a) Prove that the substitution u = yl¬n reduces the Bemoulli's equation
1-n
+ P(x)y = f(x)y", n ± 0 , n ± 1
dx
to a linear equation in u.
(b) Find the solution of the nonlinear IVP
dy
x2
- 2xy
dx
3y³,
y(1) = 1.

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