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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 initial value problem in f''(t)+9f(t)=0;f(0)=0,f(ττ2)=1

The solution is. ft=sin3t

See the step by step solution

Step by Step Solution

Step 1: Definition of characteristic polynomial

Consider the linear differential operator


The characteristic polynomial of T is defined as


Step 2: Determination of the solution

The characteristic polynomial of the operator as follows.


Then the characteristic polynomial is as follows.

PT λ=λ2+9

Solve the characteristic polynomial and find the roots as follows.

λ2+9=0 λ2=9 λ=9 λ=±3i

Therefore, the roots of the characteristic polynomials are 3i and 3i.

Step 3: Explanation of the solution

Since, the roots of the characteristic equation are different complex numbers.

The exponential functions e3it and e-3it form a basis of the kernel of T .

Hence, they form a basis of the solution space of the homogenous differential equation is Tf=0.

Thus, the general solution of the differential equation f''t+9f't=0 is ft=c1e3it+c2e-3it .

Step 4: Explanation of the solution with initial value problem

Consider the general solution of the f''t+9f't=0 with the initial value problem f0=0,fπ2=1 is as follows,


Simplify further,


Hence the solution is ft=-sin3t.


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