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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' = 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 +...

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 15CR
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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 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 + ...
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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