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Linear Algebra With Applications
Found in: Page 442
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

Solve the differential equation d2xdt2+2x=cos(t) and find all the real solutions of the differential equation.

The solution is xt=c1cos2t+c2etsin2t+cost .

See the step by step solution

Step by Step Solution

Step1: Definition of characteristic polynomial

Consider the linear differential operator


The characteristic polynomial of is defined as


Step2: Determination of the solution

The characteristic polynomial of the operator as follows.


Then the characteristic polynomial is as follows.


Solve the characteristic polynomial and find the roots as follows.

λ2+2=0 λ2=-2 λ=±2i

Therefore, the roots of the characteristic polynomials are 2i and -2i .

Step3: Simplification of the solution

Since, the roots of the characteristic equation is different real numbers.

The exponential functions e2it and e-2it form a basis of the kernel of T .

Hence, they form a basis of the solution space of the homogenous differential equation is T (f) = 0 .

The complementary function as follows.


Step4: To find the particular solution

The particular solution to the differential equation x"+2x+=cos(t) is of the form role="math" localid="1660806105789" xpt=Acost+Bsint .

Differentiate the function xp(t)=Acost+Bsint with respect to t as follows.

xpt=Acost+Bsint x'=-Asin+Bcost

Similarly, differentiate the function x'=-Asint+Bcost with respect to t as follows.


Substitute the values -Acost-Bsint for x" and Acost+Bsint for x in x"+2x as follows.

x"+2x=-Acost-Bsint+2Acost+Bsint =-Acost-Bsint+2Acost+2Bsint =Acost+Bsintx"+2x=Acost+Bsint

Substitute the value Acost+Bsint for x"+2x in x"+2x = cos(t) as follows.


Compare the coefficients of Cost and sint as follows.

A = 1

B = 0

Substitute the value 1 for A and 0 for B in role="math" localid="1660806778527" xp(t)=Acos(t)+Bsin(t) as follows.


Therefore, the general solution as follows.


Thus, the general solution of the differential equation x"+2x=cost is xt=c1cos2t+c2etsin2t+cost .

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