**Problem 4:** Solve the initial value problem given by$x^3 y''' + xy' - y = 0,$ $y(1) = 1, \quad y'(1) = 0, \quad y''(1) = 1$

Answered: x°y "у" , у(1) — 0, у"(1) — 1 + ху — у…

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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**Problem 4:** Solve the initial value problem given by

$x^3 y''' + xy' - y = 0,$

$y(1) = 1, \quad y'(1) = 0, \quad y''(1) = 1$

Expert Solution

Step 1

$x^{3} y "' + x y' - y = 0 \Rightarrow D (D - 1) (D - 2) + D - 1 = 0 w h e r e w e c o n s i d e r z = \log x o n s o l v i n g c u b i c e q u a t i o n w e g e t D^{3} - 3 D^{2} + 3 D - 1 = 0 {(D - 1)}^{3} = 0 \Rightarrow D = 1, 1, 1 y (z) = (c_{1} + c_{2} z + c_{3} z^{2}) e^{z} y (x) = (c_{1} + c_{2} \log x + c_{3} {(\log x)}^{2}) x u s e i n t i a l c o n d i t i o n t o f i n d c_{1} = 1, c_{2} = - 1, c_{3} = 1 y (x) = (1 - \log x + {(\log x)}^{2}) x$

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