Follow the steps below to solve the given differential equation using series methods. Assuming the solution can be represented by a power series y' = a) Find the first and second derivatives of y. y' = Σ n=2 n=0 Σ n=1 b) Substituting y, y', y'' into the equation gives (−2+ )’’+(1+x)y’+5y=0, y(0) = 3, y’(0) = 1 an+ 2 = where: ao = a1 = ∞ Σ n=2 a2 = c) After shifting the summation indices to start from the same values and have the same exponent of a, combine the summations into a single summation. a3 = + a4 = ∞ y = Σ anxn n=0 ∞ n=1 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. + Σ n=0 an +1 + e) Using the initial values and the recursive formula, determine the first few terms of the series solution x = 0 = 0 an y = a + α₁x + ₂x² + a3x³ + α₁x² +...
Follow the steps below to solve the given differential equation using series methods. Assuming the solution can be represented by a power series y' = a) Find the first and second derivatives of y. y' = Σ n=2 n=0 Σ n=1 b) Substituting y, y', y'' into the equation gives (−2+ )’’+(1+x)y’+5y=0, y(0) = 3, y’(0) = 1 an+ 2 = where: ao = a1 = ∞ Σ n=2 a2 = c) After shifting the summation indices to start from the same values and have the same exponent of a, combine the summations into a single summation. a3 = + a4 = ∞ y = Σ anxn n=0 ∞ n=1 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. + Σ n=0 an +1 + e) Using the initial values and the recursive formula, determine the first few terms of the series solution x = 0 = 0 an y = a + α₁x + ₂x² + a3x³ + α₁x² +...
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 below to solve the given differential equation using series methods.
Assuming the solution can be represented by a power series
Y Σ
n=1
∞
y'' = Σ
n=2
a) Find the first and second derivatives of y.
∞
n=2
b) Substituting y, y', y'' into the equation gives
∞
Σ
n=0
an + 2 =
where:
ao
2+x)’+(1+x)y’+5y=0, g(0) = 3, y’(0) =1
||
a1 =
a2
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 =
∞
+Σ
a3 =
∞
Σ
n=0
n=1
a4 =
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.
anxn
+
an +1 +
e) Using the initial values and the recursive formula, determine the first few terms of the
series solution
∞
n=0
x = 0
= 0
an
y = a + α₁x + ²x² + α³x³ + α₁x¹ + ...
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