In this exercise we consider finding the first five coefficients in the series solution of the first order linear initial value problem 3y" - xy + 4y = 0 subject to the initial condition y(0) = 1./ (0) = 3. Since the equation has an ordinary point at x = 0 and it has a power series solution in the form We learned how to easily solve problems like this separation of variables but here we want to consider the power series method. (1) Insert the formal power series into the differential equation and derive the recurrence relation Cn-2 for n=2, 3,... The solution to this initial value problem can be written in the form y(x) = Coyi (2) + C₁2(x) where co and c₁ are determined from the initial conditions. The function y(x) is an even function and y/₂() is an odd function. For this example, from the initial conditions, we have co = and C₁ = The function 3/2 (2) is an infinite series ₂(x) = x + a₂+1²+1 (2) Use the recurrence relation to find the first few coefficients of the infinite series a3 as y=[c₂x" n-0 a7 The function 3₁ (2) is an even degree polynomial y₁= other words, note that the constant co has been factored out. NOTE note that the constant C₁ has been factored out. NOTE In the function y₁ (2) the first term is 1. In
In this exercise we consider finding the first five coefficients in the series solution of the first order linear initial value problem 3y" - xy + 4y = 0 subject to the initial condition y(0) = 1./ (0) = 3. Since the equation has an ordinary point at x = 0 and it has a power series solution in the form We learned how to easily solve problems like this separation of variables but here we want to consider the power series method. (1) Insert the formal power series into the differential equation and derive the recurrence relation Cn-2 for n=2, 3,... The solution to this initial value problem can be written in the form y(x) = Coyi (2) + C₁2(x) where co and c₁ are determined from the initial conditions. The function y(x) is an even function and y/₂() is an odd function. For this example, from the initial conditions, we have co = and C₁ = The function 3/2 (2) is an infinite series ₂(x) = x + a₂+1²+1 (2) Use the recurrence relation to find the first few coefficients of the infinite series a3 as y=[c₂x" n-0 a7 The function 3₁ (2) is an even degree polynomial y₁= other words, note that the constant co has been factored out. NOTE note that the constant C₁ has been factored out. NOTE In the function y₁ (2) the first term is 1. In
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:In this exercise we consider finding the first five coefficients in the series solution of the first order linear initial value problem
3y" - xy' + 4y =0 subject to the initial condition y(0) = 1. (0) = 3.
Since the equation has an ordinary point at x = 0 and it has a power series solution in the form
We learned how to easily solve problems like this separation of variables but here we want to consider the power series method.
(1) Insert the formal power series into the differential equation and derive the recurrence relation
Cn-2 for n=2, 3,...
The solution to this initial value problem can be written in the form y(z) = Coyi (z) + C₁2(x) where co and c₁ are determined from the initial
conditions. The function ₁ (2) is an even function and y/₂ (2) is an odd function.
For this example, from the initial conditions, we have co
and C₁ =
The function 3/₂ (2) is an infinite series y₂(x) = x +
(2) Use the recurrence relation to find the first few coefficients of the infinite series
a3
as
y=[c₁z"
n-0
a2+12+1
a7
The function 3₁ (a) is an even degree polynomial y₁=
other words, note that the constant co has been factored out.
NOTE note that the constant C₁ has been factored out.
NOTE In the function ₁ (z) the first term is 1. In
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