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" = n=2 = n=0 n=1 b) Substituting y, y', y'' into the equation gives where: M8 an +2 = ( − 5 + x)y'' + (1 − 2x)y' + y = 0, y(0) = 3, y'(0) = 1 ao 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. n=2 |||| a1 = ∞ y = 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. n=1 || az = Anxn e) Using the initial values and the recursive formula, determine the first few terms of the series solution n=0 an +1 + x = 0 - 0 an y = a + a₁x + ª²x² + α3x³ + a²x² +
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" = n=2 = n=0 n=1 b) Substituting y, y', y'' into the equation gives where: M8 an +2 = ( − 5 + x)y'' + (1 − 2x)y' + y = 0, y(0) = 3, y'(0) = 1 ao 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. n=2 |||| a1 = ∞ y = 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. n=1 || az = Anxn e) Using the initial values and the recursive formula, determine the first few terms of the series solution n=0 an +1 + x = 0 - 0 an y = a + a₁x + ª²x² + α3x³ + a²x² +
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' =
y"
=
a) Find the first and second derivatives of y.
n=2
n=0
( − 5 + x)y'' + (1 − 2x)y' + y = 0, y(0) = 3, y'(0) = 1
8
Σ
n=1
an +2 =
M8
where:
n=2
b) Substituting y, y', y'' into the equation gives
ao =
a1 =
a2
=
=
∞
+Σ
a3
a4 =
||
∞
n=1
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.
anxn
+
an +1 +
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.
xn
=
0
e) Using the initial values and the recursive formula, determine the first few terms of the
series solution
= 0
An
y = a + a₁x + ²x² + A³x³ + α²x¹ + ...
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