Find y as a function of t if1296y" + 72y' + y = 0,y(0) = 4, 3'(0) = 7.y =

Answered: Find y as a function of t if 1296y" +…

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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Find y as a function of t if
1296y" + 72y' + y = 0,
y(0) = 4, 3'(0) = 7.
y =

Expert Solution

Step 1

The given equation is second order homogeneous ODE has the form of ay’’ + by’ + cy = 0

The nature of roots is given as:

$\begin{matrix} Corresponding part of C.F. & Nature of roots of auxiliary equation \\ 1 & One real root m_{1} & c_{1} e^{m_{1} t} \\ 2 & Two real and different root m_{1} and m_{2} & c_{1} e^{m_{1} t} + c_{2} e^{m_{2} t} \\ 3 & Three real and different root m_{1}, m_{2} and m_{3} & c_{1} e^{m_{1} t} + c_{2} e^{m_{2} t} + c_{3} e^{m_{3} t} \\ 4 & \begin{array}{l} Two real and equal roots \\ m_{1}, m_{1} \end{array} & (c_{1} + t c_{2}) e^{m_{1} t} \\ 5 & \begin{array}{l} Three real and equal roots \\ m_{1}, m_{1}, m_{1} \end{array} & (c_{1} + t c_{2} + t^{2} c_{3}) e^{m_{1} t} \\ 6 & One pair of complex root α \pm β i & e^{α t} (c_{1} \cos β t + c_{2} \sin β t) \end{matrix}$

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