Follow the steps to solve the below differential equation using series methods. Assuming the solution can be represented by a power series ·Lª n=0 y' a) Find the first and second derivatives of y. = y'' = Σ n=2 ∞ Σ n=1 Σ n=0 8 b) Substituting y, y', y'' into the equation gives an+2= where: y'' + 4xy' - 2y = 0, y(0) = 1, y'(0) = 3 ao = Σ n=2 a₁ = 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. a2 = az = y = 8 a4 = +Σ 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. anx e) Using the initial values and the recursive formula, determine the first few terms of the series solution ∞ + Σ n=0 0 = 0 y = a + ₁x + a²x² + α³x³ + α₁x¹ + ...
Follow the steps to solve the below differential equation using series methods. Assuming the solution can be represented by a power series ·Lª n=0 y' a) Find the first and second derivatives of y. = y'' = Σ n=2 ∞ Σ n=1 Σ n=0 8 b) Substituting y, y', y'' into the equation gives an+2= where: y'' + 4xy' - 2y = 0, y(0) = 1, y'(0) = 3 ao = Σ n=2 a₁ = 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. a2 = az = y = 8 a4 = +Σ 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. anx e) Using the initial values and the recursive formula, determine the first few terms of the series solution ∞ + Σ n=0 0 = 0 y = a + ₁x + a²x² + α³x³ + α₁x¹ + ...
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
·Lª
n=0
y'
a) Find the first and second derivatives of y.
=
y'' =
Σ
n=2
∞
Σ
n=1
Σ
n=0
8
b) Substituting y, y', y'' into the equation gives
an+2=
where:
y'' + 4xy' - 2y = 0, y(0) = 1, y'(0) = 3
ao =
Σ
n=2
a₁ =
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.
a2 =
az =
y =
8
a4 =
+Σ
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.
anx
e) Using the initial values and the recursive formula, determine the first few terms of the
series solution
∞
+ Σ
n=0
0
= 0
y = a + ₁x + ²x² + α³x³ + α₁x¹ + ...
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