Follow the steps to solve the below differential equation using series methods. Assuming the solution can be represented by a power series ∞ y' = Σ n=1 y' = a) Find the first and second derivatives of y. Σ n=2 y'' + xy' + 4y = 0, y(0) = 3, y'(0) = 1 ∞ Σ n=2 n=0 b) Substituting y, y', y'' into the equation gives an+ 2 = y = ∞ +Σ n=0 n=1 anxen + ∞ n=0 c) After shifting the summation indices to start from the same values and have the same exponent of x, combine the summations into a single summation. = 0 x = 0 d) Given that if a power series is zero for all x, all its coefficients must be zero, find a recursive formula for the solution. e) Using the initial values and the recursive formula, determine the first few terms of the series solution
Follow the steps to solve the below differential equation using series methods. Assuming the solution can be represented by a power series ∞ y' = Σ n=1 y' = a) Find the first and second derivatives of y. Σ n=2 y'' + xy' + 4y = 0, y(0) = 3, y'(0) = 1 ∞ Σ n=2 n=0 b) Substituting y, y', y'' into the equation gives an+ 2 = y = ∞ +Σ n=0 n=1 anxen + ∞ n=0 c) After shifting the summation indices to start from the same values and have the same exponent of x, combine the summations into a single summation. = 0 x = 0 d) Given that if a power series is zero for all x, all its coefficients must be zero, find a recursive formula for the solution. e) Using the initial values and the recursive formula, determine the first few terms of the series solution
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:Follow the steps to solve the below differential equation using series methods.
Assuming the solution can be represented by a power series
y' = Σ
y"
a) Find the first and second derivatives of y.
n=2
n=0
IM8 IM8
n=1
b) Substituting y, y', y'' into the equation gives
+Σ
an+ 2 =
where:
a2
ao
a1 =
a3
a4
y'' + xy' + 4y = 0, y(0) = 3, y'(0) = 1
c) After shifting the summation indices to start from the same values and have the same
exponent of x, combine the summations into a single summation.
||
Y
||
∞
n=1
||
n=0
||
Anxn
d) Given that if a power series is zero for all x, all its coefficients must be zero, find a
recursive formula for the solution.
+
e) Using the initial values and the recursive formula, determine the first few terms of the
series solution
M8
n=0
xn
-
0
= 0
y = a + a₁x + a₂x² + α3x³ + α²x¹ + ...
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