1- Explain the the difference between DE with Ordinary points and DE with singular points Find two power series solutions of the given differential equation about the ordinary point x = 0. y" - 4xy + y = 0 We are asked to find two power series solutions to the following homogenous linear second-order differential equation. y" – 4xy' + y = 0 By Theorem 6.2.1, we know two such solutions exist about the ordinary point x= 0. In fact, the differential equation has no singular points as each of the coefficient functions are (very simple) polynomials. Therefore, they are trivially represented as power series at any point and are therefore analytic. This means the power series solution will converge for Ixl < o. Let y = . We substitute this series and its derivatives into the given differential equation. We will n=0 solve for two sets of cn to find the two solutions. The differential equation includes non-zero terms for the first and second derivative of y. These derivatives always have the same form, so we refer to equation (1) of section 6.1. Y = C,nx" -1 n-1 E ,m(n - 1)x - 2 n- 2 Substitute the power series into the given differential equation. y" - 4xy + y = - 4

1- Explain the the difference between DE with Ordinary points and DE with singular points Find two power series solutions of the given differential equation about the ordinary point x = 0. y" - 4xy + y = 0 We are asked to find two power series solutions to the following homogenous linear second-order differential equation. y" – 4xy' + y = 0 By Theorem 6.2.1, we know two such solutions exist about the ordinary point x= 0. In fact, the differential equation has no singular points as each of the coefficient functions are (very simple) polynomials. Therefore, they are trivially represented as power series at any point and are therefore analytic. This means the power series solution will converge for Ixl < o. Let y = . We substitute this series and its derivatives into the given differential equation. We will n=0 solve for two sets of cn to find the two solutions. The differential equation includes non-zero terms for the first and second derivative of y. These derivatives always have the same form, so we refer to equation (1) of section 6.1. Y = C,nx" -1 n-1 E ,m(n - 1)x - 2 n- 2 Substitute the power series into the given differential equation. y" - 4xy + y = - 4

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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Question

1- Explain the the difference between DE with Ordinary points and DE with singular points
Find two power series solutions of the given differential equation about the ordinary point x = 0o.
y" - 4xy' + y = 0
We are asked to find two power series solutions to the following homogenous linear second-order differential
equation.
y" - 4xy' + y = 0
By Theorem 6.2.1, we know two such solutions exist about the ordinary point x = 0. In fact, the differential
equation has no singular points as each of the coefficient functions are (very simple) polynomials. Therefore,
they are trivially represented as power series at any point and are therefore analytic. This means the power
series solution will converge for |x| < oo.
Let y =
5C". We substitute this series and its derivatives into the given differential equation. We will
n = 0
solve for two sets of cn to find the two solutions.
The differential equation includes non-zero terms for the first and second derivative of y. These derivatives
always have the same form, so we refer to equation (1) of section 6.1.
Y = ) C,nxn - 1
n = 1
y" =
E on(n – 1)x" - 2
n = 2
Substitute the power series into the given differential equation.
y" - 4xy' + y =
- 4
-
n = 1
n = 0

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