Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question

Solve for the general solution for the differential equation:

 

(a). 2t²u" + 3tu' - u = 0,t> 0
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Transcribed Image Text:(a). 2t²u" + 3tu' - u = 0,t> 0
Expert Solution
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Step 1: given-:

The given differential equation is,

2 t squared u apostrophe apostrophe plus 3 t u apostrophe minus u equals 0 o r comma 2 t squared fraction numerator d squared u over denominator d t squared end fraction plus 3 t fraction numerator d u over denominator d t end fraction minus u equals 0

let us substitute t=ez so z=logt and hence

 t fraction numerator d u over denominator d t end fraction equals fraction numerator d u over denominator d z end fraction space a n d t squared fraction numerator d squared u over denominator d t squared end fraction equals fraction numerator d u over denominator d z end fraction open parentheses fraction numerator d u over denominator d z end fraction minus u close parentheses equals fraction numerator d over denominator d z end fraction open parentheses fraction numerator d over denominator d z end fraction minus 1 close parentheses u l e t space u s space c o n s i d e r space D equals fraction numerator d over denominator d z end fraction rightwards double arrow t fraction numerator d u over denominator d t end fraction equals fraction numerator d u over denominator d z end fraction equals D u space a n d t squared fraction numerator d squared u over denominator d t squared end fraction equals fraction numerator d u over denominator d z end fraction open parentheses fraction numerator d u over denominator d z end fraction minus u close parentheses equals fraction numerator d over denominator d z end fraction open parentheses fraction numerator d over denominator d z end fraction minus 1 close parentheses u equals D open parentheses D minus 1 close parentheses u s o comma a u x i l l i a r y space e q u a t i o n space i s space g i v e n space b y comma open parentheses 2 D open parentheses D minus 1 close parentheses plus 3 D minus 1 close parentheses u equals 0 rightwards double arrow 2 D squared minus 2 D plus 3 D minus 1 equals 0 rightwards double arrow 2 D squared plus D minus 1 equals 0 rightwards double arrow 2 D squared plus 2 D minus D minus 1 equals 0 rightwards double arrow 2 D open parentheses D plus 1 close parentheses minus 1 open parentheses D plus 1 close parentheses equals 0 rightwards double arrow open parentheses 2 D minus 1 close parentheses open parentheses D plus 1 close parentheses equals 0 rightwards double arrow D equals 1 half comma negative 1

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