For each of the following differential equations: ODE1. (2 + x)y" + (1+x)y' + 3y = 0 ODE2. x?y" + x(1+ x)y' – 3(3+x)y = 0 ODE3. x²(1+ x)y" – x(3 – x)y' + 4y = 0 Determine if xo = 0 is an ordinary or a singular point. If it is a singular point, determine if it is a regular or an irregular singular point. 1. 2. Based on your results in (a), use the appropriate method to determine two linearly independent series solutions about x, = 0. Indicate, the indicial equation, the root(s) of the indicial equation, and the recurrence relation, where applicable. а. For ODE2, use formula (4.17) to determine its second series solution. Hint: substitute series for e-x b. For ODE3, verify that the first series solution can be written as: Y1 = -Σ-1)" (n + 1)*χn*2 In=0 Then, use formula (4.15) or (4.16) to determine its second series solution. 3. For all three equations, write the first four non-zero terms (unless it terminates earlier) of each series solutions, where relevant.
For each of the following differential equations: ODE1. (2 + x)y" + (1+x)y' + 3y = 0 ODE2. x?y" + x(1+ x)y' – 3(3+x)y = 0 ODE3. x²(1+ x)y" – x(3 – x)y' + 4y = 0 Determine if xo = 0 is an ordinary or a singular point. If it is a singular point, determine if it is a regular or an irregular singular point. 1. 2. Based on your results in (a), use the appropriate method to determine two linearly independent series solutions about x, = 0. Indicate, the indicial equation, the root(s) of the indicial equation, and the recurrence relation, where applicable. а. For ODE2, use formula (4.17) to determine its second series solution. Hint: substitute series for e-x b. For ODE3, verify that the first series solution can be written as: Y1 = -Σ-1)" (n + 1)*χn*2 In=0 Then, use formula (4.15) or (4.16) to determine its second series solution. 3. For all three equations, write the first four non-zero terms (unless it terminates earlier) of each series solutions, where relevant.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Solve ODE3 question 2b, 3
Transcribed Image Text:Cave 1: if r-r is nut
a zero, nut a portive intege, then there
Px(sts the seond seres sol
of the form :
Y. (x) =
, b, x"
(4.4)
bo to
Case 2: if rFOie. erL
the seond suls is
(x) = Y. (x) In 1x + , An x^*
(4.15)
always contains lopanthmic term?
N is
Here :
secund sul
Case 3: - N
po stthve integer then second so1=
where
a
of the forma;
20 or nun Zero.
Y (x) = ky, tx) In /x) + E Ba xtri
(4:16)
B. to. Constant k may be 0, if u,
sol
are as in Case l.
If kt0
contains
term.
secund sol-
Transcribed Image Text:For each of the following differential equations:
ODE1.
(2 + x)y" + (1 + x)y' + 3y = 0
ODE2.
х?у" + x(1 + х)у' — 3(3 + х)у %3
= 0
ODE3.
х?(1 + х)у" — x(3 — х)у' + 4у %3D 0
=
Determine if xo = 0 is an ordinary or a singular point. If it is a singular point, determine if it
is a regular or an irregular singular point.
1.
2.
Based on your results in (a), use the appropriate method to determine two linearly
independent series solutions about x, = 0. Indicate, the indicial equation, the root(s) of the
indicial equation, and the recurrence relation, where applicable.
For ODE2, use formula (4.17) to determine its second series solution.
a.
Hint: substitute series for e-*
b.
For ODE3, verify that the first series solution can be written as:
00
Y1 =
- -1)" (n + 1)%χ"+2
In=0
Then, use formula (4.15) or (4.16) to determine its second series solution.
For all three equations, write the first four non-zero terms (unless it terminates earlier) of
each series solutions, where relevant.
3.
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