Series Solution Suppose we are given the initial value problem: with y(ro) form Problem: a₂(x)y" + a₁(x)y' + ao(x)y=0 = a and y'(xo) = 3. We assume the solution can be written in the power series ∞ By the formula an = (o) and the initial conditions, we may find the values of the starting coefficients. By plugging the series expansion back in the equation, we may find a recurrence equation for the {an}. Based on these, we may find a solution y in the series form. y = Σan (x-2 xo)". n=0 Steps Consider the following initial value problem: x²y" + xy + x²y = 0 with y(0) = 1 and y'(0) = 0. The center of the power series is xo = 0. 1. Use the initial conditions to find ao and a₁ by hand (the first two coefficients of the series solution). Print out ao and a₁ in Python. 1 2. Use the series assumption to plug in the equation and find the recurrence equation by hand. Print the recurrence equation in Python. 3. By the recurrence equation and the values of ao and a₁, use a loop argument in Python
Series Solution Suppose we are given the initial value problem: with y(ro) form Problem: a₂(x)y" + a₁(x)y' + ao(x)y=0 = a and y'(xo) = 3. We assume the solution can be written in the power series ∞ By the formula an = (o) and the initial conditions, we may find the values of the starting coefficients. By plugging the series expansion back in the equation, we may find a recurrence equation for the {an}. Based on these, we may find a solution y in the series form. y = Σan (x-2 xo)". n=0 Steps Consider the following initial value problem: x²y" + xy + x²y = 0 with y(0) = 1 and y'(0) = 0. The center of the power series is xo = 0. 1. Use the initial conditions to find ao and a₁ by hand (the first two coefficients of the series solution). Print out ao and a₁ in Python. 1 2. Use the series assumption to plug in the equation and find the recurrence equation by hand. Print the recurrence equation in Python. 3. By the recurrence equation and the values of ao and a₁, use a loop argument in Python
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please solve the parts that say “by hand”
No programming needed
Transcribed Image Text:Series Solution
Suppose we are given the initial value problem:
a2(x)y" + a₁(x)y' + ao(x)y=0
with y(ro) = a and y'(xo) = ß. We assume the solution can be written in the power series
form
y = Σan(x-xo)".
By the formula an = ((o) and the initial conditions, we may find the values of the starting
coefficients. By plugging the series expansion back in the equation, we may find a recurrence
equation for the {an}. Based on these, we may find a solution y in the series form.
Problem:
8
n=0
Consider the following initial value problem:
x²y + xy + x²y = 0
with y(0) = 1 and y'(0) = 0. The center of the power series is o = 0.
Steps
1. Use the initial conditions to find ao and a₁ by hand (the first two coefficients of the
series solution). Print out ao and a, in Python.
1
2. Use the series assumption to plug in the equation and find the recurrence equation by
hand. Print the recurrence equation in Python.
3. By the recurrence equation and the values of ao and a₁, use a loop argument in Python
to find a2,..., a10. Print your answer in Python.
4. Can you find the general formula for an by hand? Print out your answer in Python. €
00
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