10.d² u1 dudx²1+xdxu(0) = 0, u(1) = 0 (n = 3)+= -1, 0 10.d² u1 dudx²1+xdxu(0) = 0, u(1) = 0 (n = 3)+= -1, 0

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Answered: 10. d² u 1 du + dx² 1+xdx u(0) = 0,…

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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Related questions

Question

10.
d² u
1 du
dx²
1+xdx
u(0) = 0, u(1) = 0 (n = 3)
+
= -1, 0<x< 1

Expert Solution

Step 1: Applying substition to start with:

$fraction numerator d squared u over denominator d x squared end fraction plus fraction numerator 1 over denominator 1 plus x end fraction fraction numerator d u over denominator d x end fraction equals negative 1 P u t t i n g space fraction numerator d u over denominator d x end fraction equals v comma space o n e space m a y space f i n d space fraction numerator d v over denominator d x end fraction plus fraction numerator v over denominator 1 plus x end fraction equals negative 1 w h i c h space i s space l i n e a r. I F equals e to the power of integral fraction numerator 1 over denominator 1 plus x end fraction d x end exponent equals left parenthesis 1 plus x right parenthesis H e n c e comma space left parenthesis 1 plus x right parenthesis fraction numerator d v over denominator d x end fraction plus v equals negative left parenthesis 1 plus x right parenthesis fraction numerator d left parenthesis left parenthesis 1 plus x right parenthesis v right parenthesis over denominator d x end fraction equals negative left parenthesis 1 plus x right parenthesis left parenthesis 1 plus x right parenthesis v equals negative x minus x squared over 2 plus c H e n c e space fraction numerator d u over denominator d x end fraction equals negative fraction numerator x over denominator 1 plus x end fraction minus 1 half fraction numerator x squared over denominator 1 plus x end fraction plus fraction numerator c over denominator 1 plus x end fraction H e n c e space u equals 1 half ln left parenthesis 1 plus x right parenthesis minus x minus 1 half open parentheses x squared over 2 minus x close parentheses plus c ln left parenthesis 1 plus x right parenthesis plus d$