6 Findal9 fuuctio n 0f8y"ry't&y=y Co) = 7y(a)= ?y co) =/y"(0)=706.ylu):

Answered: Find a function 0f nH al =0 y Co) = 7 y…

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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Question

Find
al
9 fuuctio n 0f
8y"ry't&y=
y Co) = 7
y(a)= ?
y co) =/
y"(0)=70
6.
ylu):

Expert Solution

Step 1

The differential equation is given as

$y''' - 8 y'' - y' + 8 y = 0$

The corresponding auxiliary equation is

$\begin{array}{rcl} m^{3} - 8 m^{2} - m + 8 & = & 0 \\ m^{2} (m - 8) - 1 (m - 8) & = & 0 \\ (m^{2} - 1) (m - 8) & = & 0 \\ (m + 1) (m - 1) (m - 8) & = & 0 \\ m & = & - 1, 1, 8 \end{array}$

Hence the complementary function is

$y = C_{1} e^{x} + C_{2} e^{- x} + C_{3} e^{8 x} (1)$

Then

$y' (x) = C_{1} e^{x} - C_{2} e^{- x} + 8 C_{3} e^{8 x} (2)$

and

$y'' (x) = C_{1} e^{x} + C_{2} e^{- x} + 64 e^{8 x} (3)$

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Find a function 0f nH al =0 y Co) = 7 y (o) =/