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 a) Find the first and second derivatives of y. y' = Σ n=2 Σ n=0 ∞ Σ n=2 b) Substituting y, y', y'' into the equation gives y'' - 5xy' − 3y = 0, y(0) = 1, y'(0) = 2 an+ 2 = where: + ao = a₁ = a2 = y = Σ n=0 ∞ az = a3 a4 = n=1 anxen c) After shifting the summation indices to start from the same values and have the sam exponent of x, combine the summations into a single summation. +Σ n=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. x = 0 = 0 e) Using the initial values and the recursive formula, determine the first few terms of th series solution = y = a +α₁x + ²x² + α3x³ + α₁x² + ...
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 a) Find the first and second derivatives of y. y' = Σ n=2 Σ n=0 ∞ Σ n=2 b) Substituting y, y', y'' into the equation gives y'' - 5xy' − 3y = 0, y(0) = 1, y'(0) = 2 an+ 2 = where: + ao = a₁ = a2 = y = Σ n=0 ∞ az = a3 a4 = n=1 anxen c) After shifting the summation indices to start from the same values and have the sam exponent of x, combine the summations into a single summation. +Σ n=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. x = 0 = 0 e) Using the initial values and the recursive formula, determine the first few terms of th series solution = y = a +α₁x + ²x² + α3x³ + 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 to solve the below differential equation using series methods.
Assuming the solution can be represented by a power series
y' =
y"
a) Find the first and second derivatives of y.
n=2
Σ
n=0
İM: İM8
b) Substituting y, y', y'' into the equation gives
An +2 =
where:
ao =
a1
a2
y'' — 5xy' — 3y = 0, y(0) = 1, y'(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.
=
=
y
+
a3
a4 =
n=0
n=1
d) Given that if a power series is zero for all o, all its coefficients must be zero, find a
recursive formula for the solution.
||
Anxn
e) Using the initial values and the recursive formula, determine the first few terms of the
series solution
n=0
x = 0
2
= 0
y = A。 + A₁x + A²x² + α3x³ + α²x² +
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