Problem #5 Solving 2nd-Order, Linear, Homogeneous ODES Given the 2nd-order, linear, homogeneous ODE y"(x) + P(x)y'(x) + Q(x)y(x) = 0, it can be shown that if the quantity Q'x) + 2P(x)Q(x) (Q(x))32 is a constant, then the transformation -S JaQx) dx = Z for any non-zero constant a will transform the original ODE to the ODE Y"(2) + -Y'(2) + 금y(2) = 0 2 Ja which has constant coefficients. The solution to the original ODE is then - [ JaQ¢x) dx. y(x) = ¥(2) with Z = Use this idea to determine a general solution to the following ODE 2xy" (x) + y'(x) + dy(x) = 0 for x > 0.
Problem #5 Solving 2nd-Order, Linear, Homogeneous ODES Given the 2nd-order, linear, homogeneous ODE y"(x) + P(x)y'(x) + Q(x)y(x) = 0, it can be shown that if the quantity Q'x) + 2P(x)Q(x) (Q(x))32 is a constant, then the transformation -S JaQx) dx = Z for any non-zero constant a will transform the original ODE to the ODE Y"(2) + -Y'(2) + 금y(2) = 0 2 Ja which has constant coefficients. The solution to the original ODE is then - [ JaQ¢x) dx. y(x) = ¥(2) with Z = Use this idea to determine a general solution to the following ODE 2xy" (x) + y'(x) + dy(x) = 0 for x > 0.
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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Topic Video
Question
The second image is problem 5 and its solution (hint for the question provided)
Transcribed Image Text:- Another Regular Sturm-Liouville Problem
Determine the eigenvalues (2,) and eigenfunctions (p„(x)) for the differential
equation
2.xp" (x) + q'(x) + 1@(x) = 0
for 0 <x < 1, along with the boundary conditions, ø(0) = 9(1) = 0. Hint: See Problem
#5 of Homework # 4.
Determine the "dot" product for which pm • P, = 0 when m # n and use this
to determine the coefficients a, if the function f(x) = x, is expanded as
A(x) = x = a,P„(x)
n-1
for 0 < x < 1.
Transcribed Image Text:Problem #5
Solving 2nd-Order, Linear, Homogeneous ODES
Given the 2nd-order, linear, homogeneous ODE
y"(x) + P(x)y'(x) + Q(x)y(x) = 0,
it can be shown that if the quantity
Q'(x) + 2P(x)Q(x)
(Q(x))32
is a constant, then the transformation
I JaQ(x) dx
z =
for any non-zero constant a will transform the original ODE to the ODE
Y"(2) +
-Y' (3) + 공Y(2) = 0
2 Ja
which has constant coefficients. The solution to the original ODE is then
J JaQ(x) dx.
y(x) = ¥(z)
with
= Z
Use this idea to determine a general solution to the following ODE
2xy" (x) + y'(x) + Ay(x) = 0
for x > 0.
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