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Follow the steps below to solve the given differential equation using series methods. Assuming the solution can be represented by a power series ( − 3 + x)y’’ + (1 − x)y’ – 4y = 0, y(0) = 1, y'(0) = 3 y" a) Find the first and second derivatives of y. y' = Σ IM8 M8 - || n=1 Σ y = n=2 n=0 anxen b) Substituting y, y', y'' into the equation gives

Follow the steps below to solve the given differential equation using series methods. Assuming the solution can be represented by a power series ( − 3 + x)y’’ + (1 − x)y’ – 4y = 0, y(0) = 1, y'(0) = 3 y" a) Find the first and second derivatives of y. y' = Σ IM8 M8 - || n=1 Σ y = n=2 n=0 anxen b) Substituting y, y', y'' into the equation gives

Advanced Engineering Mathematics
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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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' = Σ a) Find the first and second derivatives of y. ∞ n=2 n=0 IM8 IM8 ( − 3 + x)y’’ + (1 − x)y’ – 4y = 0, y(0) = 1, y'(0) = 3 b) Substituting y, y', y'' into the equation gives an +2 = where: n=2 ao a2 a3 a4 || a1 = + y = ∞ || M8 n=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. 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. Anxn + An +1 + n=0 e) Using the initial values and the recursive formula, determine the first few terms of the series solution x = 0 = 0 An y = a + a₁x + ª²x² + α3x³ + α²x² + ...
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' = Σ a) Find the first and second derivatives of y. ∞ n=2 n=0 IM8 IM8 ( − 3 + x)y’’ + (1 − x)y’ – 4y = 0, y(0) = 1, y'(0) = 3 b) Substituting y, y', y'' into the equation gives an +2 = where: n=2 ao a2 a3 a4 || a1 = + y = ∞ || M8 n=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. 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. Anxn + An +1 + n=0 e) Using the initial values and the recursive formula, determine the first few terms of the series solution x = 0 = 0 An y = a + a₁x + ª²x² + α3x³ + α²x² + ...
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