Series solution of variable-coefficient ODE Consider the variable coefficient linear second order homogeneous ODE (x² + 1)y" Ary' + 6y=0. 1. The point x = 0 is an ordinary point of equation (1). Therefore, we can find a power series solution of the form. 8 Y= Σ TH=0 Write down the first and second derivatives of the power series. [] 2. Substitute the power series (and its derivatives) into equatics (1). Express your answer in the form ∞ Σbm 2 + Σ¤m.T™ = 0, m=0 TL=0 where b and Cm are to be written in terms of m and am 3. Shift the index on one of the series you found in 2 so that the exponents of rare equal to m in both series. 4. Find a recurrence relation for the coefficients am 12 in terms of am and m. 15. Use the recurrence relation to find expressions for the coefficients a2, 03, 04 and 5. 6. Write down the general solution to (1) in the form y = aof(x) + aig(x). 7. Find the particular solution of (1) that satisfies the initial conditions y(0) = 3 and y'(0) = 2.
Series solution of variable-coefficient ODE Consider the variable coefficient linear second order homogeneous ODE (x² + 1)y" Ary' + 6y=0. 1. The point x = 0 is an ordinary point of equation (1). Therefore, we can find a power series solution of the form. 8 Y= Σ TH=0 Write down the first and second derivatives of the power series. [] 2. Substitute the power series (and its derivatives) into equatics (1). Express your answer in the form ∞ Σbm 2 + Σ¤m.T™ = 0, m=0 TL=0 where b and Cm are to be written in terms of m and am 3. Shift the index on one of the series you found in 2 so that the exponents of rare equal to m in both series. 4. Find a recurrence relation for the coefficients am 12 in terms of am and m. 15. Use the recurrence relation to find expressions for the coefficients a2, 03, 04 and 5. 6. Write down the general solution to (1) in the form y = aof(x) + aig(x). 7. Find the particular solution of (1) that satisfies the initial conditions y(0) = 3 and y'(0) = 2.
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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![Series solution of variable-coefficient ODE
Consider the variable coefficient linear second order homogeneous ODE
(r² + 1)y" - Ary' + 6y = 0.
1. The point r = 0 is an ordinary point of equation (1). Therefore, we can find a power series
solution of the form.
8
Y=
Σ
711=0)
Write down the first and second derivatives of the power series.
[] 2. Substitute the power series (and its derivatives) into equatics (1). Express your answer in
the form
8
Σmam 2
Σ
<=0,
m=0
TIL 0
where bin and Cm are to be written in terms of m and am.
F] 3. Shift the index on one of the series you found in 2 so that the exponents of r are equal to
m in both series.
4. Find a recurrence relation for the coefficients am 12 in terms of am and m.
5. Use the recurrence relation to find expressions for the coefficients a2, a3, a4 and a5.
6. Write down the general solution to (1) in the form y = aof(x) + a1g(x).
7. Find the particular solution of (1) that satisfies the initial conditions y(0) = 3 and y'(0) = 2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feefaa88d-d6f2-453f-b720-d6bcd4d513fe%2F0a72da51-2865-454f-870a-95af9278d4d8%2Fiirdm5r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Series solution of variable-coefficient ODE
Consider the variable coefficient linear second order homogeneous ODE
(r² + 1)y" - Ary' + 6y = 0.
1. The point r = 0 is an ordinary point of equation (1). Therefore, we can find a power series
solution of the form.
8
Y=
Σ
711=0)
Write down the first and second derivatives of the power series.
[] 2. Substitute the power series (and its derivatives) into equatics (1). Express your answer in
the form
8
Σmam 2
Σ
<=0,
m=0
TIL 0
where bin and Cm are to be written in terms of m and am.
F] 3. Shift the index on one of the series you found in 2 so that the exponents of r are equal to
m in both series.
4. Find a recurrence relation for the coefficients am 12 in terms of am and m.
5. Use the recurrence relation to find expressions for the coefficients a2, a3, a4 and a5.
6. Write down the general solution to (1) in the form y = aof(x) + a1g(x).
7. Find the particular solution of (1) that satisfies the initial conditions y(0) = 3 and y'(0) = 2.
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