1. Consider the following ODE over the interval (0, +):xy" – y' + 4x°y = 0(1)(a) Show that y = sin(x²) is a solution of ODE (1).(b) Find the general solution of ODE (1). 2. Consider the following ODE:dy(1+4e*)dx?dy+ 3e2"y = e2(x+e*)(2)dx(a) Show that if z = e", then ODE (2) is equivalent to the following ODEdydy- 4+ 3y = e2=dz(3)dz?(b) Solve ODE (3) and give the general solution of ODE (2).

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Answered: 1. Consider the following ODE over the…

1. Consider the following ODE over the interval (0,+∞): xy" – y' + 4x°y = 0 (a) Show that y = sin(x²) is a solution of ODE (1). (b) Find the general solution of ODE (1).

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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Question

1. Consider the following ODE over the interval (0, +):
xy" – y' + 4x°y = 0
(1)
(a) Show that y = sin(x²) is a solution of ODE (1).
(b) Find the general solution of ODE (1).

2. Consider the following ODE:
dy
(1+4e*)
dx?
dy
+ 3e2"y = e2(x+e*)
(2)
dx
(a) Show that if z = e", then ODE (2) is equivalent to the following ODE
dy
dy
- 4
+ 3y = e2=
dz
(3)
dz?
(b) Solve ODE (3) and give the general solution of ODE (2).

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