In the following problem apply the eigenvalue method to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. use a computer system or graphingcalculator to construct a direction field and typical solutioncurves for the given system. x'1 = 9x1 + 5x2, x'2 = -6x1 - 2x2 ; x1(0) =1,  x2(0)=0

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Chapter2: Second-order Linear Odes
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In the following problem apply the eigenvalue method to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. use a computer system or graphingcalculator to construct a direction field and typical solutioncurves for the given system.

x'1 = 9x1 + 5x2, x'2 = -6x1 - 2x; x1(0) =1,  x2(0)=0

Expert Solution
Step 1

We have to find the particular solution of the given system of differential equations. First we have

to find the general solution.

The matrix form of the given system is x1'x2'=95-6-2x1x2. Thus, the coefficient matrix is

A=95-6-2.

Step 2

Consider the matrix A-λI=9-λ5-6-2-λ. The determinant of this matrix is λ-4λ-3. Solving

the equation λ-4λ-3=0, we get, λ1=4 and λ2=3. These are the eigenvalues.

For λ=4, we have, A-λI=55-6-6. The null space of this matrix is -11. It is an eigenvector

corresponding to the eigenvalue 4.

For λ=3, we have, A-λI=65-6-5. The null space of this matrix is -561. It is an eigenvector

corresponding to the eigenvalue 3.

Let C=-11 and D=-561.

Step 3

The general solution is given by X=αCeλ1t+βDeλ2t where X=x1tx2t. So we get,

X=αCeλ1t+βDeλ2tx1tx2t=α-11e4t+β-561e3tx1tx2t=-αe4t-5β6e3tαe4t+βe3t

It is given that x10=1, x20=0. So we get,

x10x20=-αe40-5β6e30αe40+βe3010=-α-5β6α+β

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