1.Solve the initial value problem(cos x)y" – x²y' + y = 0;y(0) = 0, y'(0) = 1using power series method. Determine sufficient terms of solution y(x) that are needed to computey(0.1) accurate to five decimal places.

Answered: Solve the initial value problem (cos…

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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1.
Solve the initial value problem
(cos x)y" – x²y' + y = 0;
y(0) = 0, y'(0) = 1
using power series method. Determine sufficient terms of solution y(x) that are needed to compute
y(0.1) accurate to five decimal places.

Expert Solution

Step 1

Given that $(\cos x) y'' - x^{2} y' + y = 0, y (0) = 0, y' (0) = 1$

The objective is to solve the differential equation and find the $y (0.1)$ accurate to five decimal places.

The expansion for cos x is,

$\cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - . . .$

Take $y = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4} + a_{5} x^{5} + . . . .$

Then the differentiate $y$ ,

$y' = a_{1} + 2 a_{2} x + 3 a_{3} x^{2} + 4 a_{4} x^{3} + 5 a_{5} x^{4} + . . .$

Then differentiate $y'$ ,

$y'' = 2 a_{2} + 6 a_{3} x + 12 a_{4} x^{2} + 20 a_{5} x^{3} + 30 a_{6} x^{4} + . . .$

Step 2

Substitute y, $y', y'', \cos x$ in $(\cos x) y'' - x^{2} y' + y = 0, y (0) = 0, y' (0) = 1$

$(1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - . .) (2 a_{2} + 6 a_{3} x + 12 a_{4} x^{2} + 20 a_{5} x^{3} + 30 a_{6} x^{4} + . . .) - x^{2} a_{1} - 2 a_{2} x^{3} - 3 a_{3} x^{4} - 4 a_{4} x^{5} - 5 a_{5} x^{5} - . . . + a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4} + a_{5} x^{5} + . . . . = 0$

It is given that $a_{0} = 0, a_{1} = 1$

By taking the constant terms,

$2 a_{2} + a_{0} = 0 a_{2} = 0$

By taking the x terms common,

$6 a_{3} + a_{1} = 0 6 a_{3} = - a_{1} a_{3} = \frac{- 1}{6}$

By taking $x^{2}$ terms,

$12 a_{4} - a_{2} - a_{1} + a_{2} = 0 12 a_{4} = a_{1} a_{4} = \frac{1}{12}$

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