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*) In this question, you will find a Taylor polynomial approximation of the solution to the differential equation: dy dar fig- N Suppose that we are looking for a solution f(x) = > a;r' to the differential equation that passes through the point (0, 5). (a) What is an? dy The differential equation requires the function y and its derivative dr Also, two polynomials are equal if and only if they have the same coefficients on power terms. (b) Use this fact to find a relationship between aj and a0- We have aj = (c) Now write az in terms of a1. Then write a, in terms of ay. We have az (d) Write az in terms of a2. Then write az in terms of ag. We have az

*) In this question, you will find a Taylor polynomial approximation of the solution to the differential equation: dy dar fig- N Suppose that we are looking for a solution f(x) = > a;r' to the differential equation that passes through the point (0, 5). (a) What is an? dy The differential equation requires the function y and its derivative dr Also, two polynomials are equal if and only if they have the same coefficients on power terms. (b) Use this fact to find a relationship between aj and a0- We have aj = (c) Now write az in terms of a1. Then write a, in terms of ay. We have az (d) Write az in terms of a2. Then write az in terms of ag. We have az

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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s) In this question, you will find a Taylor polynomial approximation of the solution to the differential equation: dy = -5y dr Suppose that we are looking for a solution f(x) = > a;x' to the differential equation that passes through the point (0, 5). (a) What is a, ? The differential equation requires the function y and its derivative Also, two polynomials are equal if and only if they have the same coefficients like power terms. (b) Use this fact to find a relationship between aj and ag. We have a, = (c) Now write az in terms of aj. Then write a, in terms of an. We have a, = Op. (d) Write az in terms of a2. Then write az in terms of ag. We have az = ag = (e) Write ag in terms of az. Then write ag in terms of ag. We have a = ag = (f) You should see a pattern: aN = -a0 = (g) Combine your answers from (a)-(e) to write a polynomial approximation of degree 4 of the solution of the the differential equation. (h) Use separation of variables to find the solution of the differential equation that passes throught the point (0, 5). (i) Find a Taylor polynomial of degree 4 centred around a = 0 of the solution that you found in (h). P1 (x) = () Consider your polynomial approximation (g) to the solution to the initial value problem. Does it match with the Taylor polynomial of the solution you found in (i)?
Transcribed Image Text:s) In this question, you will find a Taylor polynomial approximation of the solution to the differential equation: dy = -5y dr Suppose that we are looking for a solution f(x) = > a;x' to the differential equation that passes through the point (0, 5). (a) What is a, ? The differential equation requires the function y and its derivative Also, two polynomials are equal if and only if they have the same coefficients like power terms. (b) Use this fact to find a relationship between aj and ag. We have a, = (c) Now write az in terms of aj. Then write a, in terms of an. We have a, = Op. (d) Write az in terms of a2. Then write az in terms of ag. We have az = ag = (e) Write ag in terms of az. Then write ag in terms of ag. We have a = ag = (f) You should see a pattern: aN = -a0 = (g) Combine your answers from (a)-(e) to write a polynomial approximation of degree 4 of the solution of the the differential equation. (h) Use separation of variables to find the solution of the differential equation that passes throught the point (0, 5). (i) Find a Taylor polynomial of degree 4 centred around a = 0 of the solution that you found in (h). P1 (x) = () Consider your polynomial approximation (g) to the solution to the initial value problem. Does it match with the Taylor polynomial of the solution you found in (i)?
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