The roots of the characterictic polynomial ofhomogeneous,constont coefficent equation as 3,3,2 thi has beengluen. According to thistlinear,a) Find the general solution of the equation6) Find the chorocteristic polynomial of the equationc)w hich order is the equation? uWrite the equation.d)If equation fl)= 1P0x+1be Equation in homogeneous withthe 2x nnhomogeneous term if converted, find the generolSolution.

Answered: The roots of the characterictic…

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

The roots of the characterictic polynomial of
homogeneous,constont coefficent equation as 3,3,2 thi has been
gluen. According to thist
linear,
a) Find the general solution of the equation
6) Find the chorocteristic polynomial of the equation
c)w hich order is the equation? uWrite the equation.
d)If equation fl)= 1P0x+1be Equation in homogeneous with
the 2x nnhomogeneous term if converted, find the generol
Solution.

Expert Solution

Step 1

It is given that the roots of the characteristic polynomial are 3, 3, 2+4i. But as we know that the complex roots always be present in pair. So, one more root of the equation will exits and it will be equal to 2-4i.

Let our equation is differential equation, which is linear and homogeneous with constant coefficients.

(a) So, general solution of the equation will be

$y = (c_{1} + c_{2} x) e^{3 x} + e^{2 x} (c_{3} \cos (4 x) + c_{4} \sin (4 x))$ .

(b) Characteristic polynomial of the equation will be

${(m - 3)}^{2} (m^{2} - 4 m + 20)$ or $m^{4} - 10 m^{3} + 53 m^{2} - 156 m + 180$ .

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