Consider the following differential equation (4-x)y"+y = 0, 0 = 0. (a) Seek a power series solution for the given differential equation about the given point vo; find the recurrence relation that the coefficients must satisfy. TU -1 ✓ On+2= an+1+ are n = 0, 1, 2,... 4 (n + 2) 4 (n+2)(n+1) (b) Find the first four nonzero terms in each of two solutions y₁ and 32. NOTE: For y, set ap = 1 and a, = 0 in the power series to find the first four - 1 in the power series to find non-zero terms. For yz. set ay = 0 and a the first four non-zero terms. 1 1 1 5 X 1(x) = 1 7,² 73 + 2²+ 25 8 96 4768 15360 1 1 1 X Y₂(x)=x 24 192 3840 (c) By evaluating the Wronskian W (y₁, 32) (zo), show that y₁ and yż form a fundamental set of solutions. 1 0 ✓ W (y1, y2) (0) = 0 1 x³ 24 2:5
Consider the following differential equation (4-x)y"+y = 0, 0 = 0. (a) Seek a power series solution for the given differential equation about the given point vo; find the recurrence relation that the coefficients must satisfy. TU -1 ✓ On+2= an+1+ are n = 0, 1, 2,... 4 (n + 2) 4 (n+2)(n+1) (b) Find the first four nonzero terms in each of two solutions y₁ and 32. NOTE: For y, set ap = 1 and a, = 0 in the power series to find the first four - 1 in the power series to find non-zero terms. For yz. set ay = 0 and a the first four non-zero terms. 1 1 1 5 X 1(x) = 1 7,² 73 + 2²+ 25 8 96 4768 15360 1 1 1 X Y₂(x)=x 24 192 3840 (c) By evaluating the Wronskian W (y₁, 32) (zo), show that y₁ and yż form a fundamental set of solutions. 1 0 ✓ W (y1, y2) (0) = 0 1 x³ 24 2:5
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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Transcribed Image Text:Consider the following differential equation
(4-x)y"+y = 0, 0 = 0.
(a) Seek a power series solution for the given differential equation
about the given point vo; find the recurrence relation that the
coefficients must satisfy.
TU
-1
✓ On+2=
a₁ +1 +
a n= 0, 1, 2,...
4 (n + 2)
4(n+2)(n+1)
(b) Find the first four nonzero terms in each of two solutions y₁ and 32.
NOTE: For y, set ap = 1 and a, = 0 in the power series to find the first four
non-zero terms. For yz. set ay = 0 and a = 1 in the power series to find
the first four non-zero terms.
1
1
1
5
X 1(x) = 1 2.²
7.3 +
2²+
25
8
96
4768
15360
1
1
1
X Y₂(x)=x
24
192
3840
(c) By evaluating the Wronskian W (y₁, 32) (zo), show that y₁ and yź
form a fundamental set of solutions.
1
0
✓ W (y1, y2) (0) =
0
1
x³
2.5
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