Solve the following system of differential equations:where&nbs

Question

$v_1 = -11 v_2 = 1 v_3 = -1$ Solve the following system of differential equations:

$frac{d_{y(t)}}{dt} = Ay(t) with y(0) =egin{bmatrix} 1 1 1 end{bmatrix}$

where

$A = egin{bmatrix} 2 & -5 & -11 0 & -2 & -9 0 & 1 & 4 end{bmatrix}$

19 Mar 2023

Answer :

Word Count : 768

We can solve this system of differential equations by finding the matrix exponential of A*t and multiplying it by the initial condition y(0):

$y(t) = e^(A*t) * y(0)$

To find the matrix exponential of At, we first need to find the eigenvalues and eigenvectors of A. We can do this by solving the characteristic equation $det(A - lambda I) = 0:$

$det(A - lambda*I) = egin{vmatrix} 2-lambda & -5 & -11 0 & -2-lambda & -9 0 & 1 & 4-lambda end{vmatrix}$

Expanding this determinant, we get:

$(2 - lambda) * egin{vmatrix} -2-lambda & -9 1 & 4-lambda end{vmatrix} - (-5) * egin{vmatrix} 0 & -9 0 & 4-lambda end{vmatrix} - (-11) * egin{vmatrix} 0 & -2-lambda 0 & 1 end{vmatrix}$

Simplifying this expression, we get:

$lambda^3 - 4lambda^2 + 3lambda = lambda * (lambda-1) * (lambda-3)$

Therefore, the eigenvalues of A are $lambda_1 = 0, lambda_2 = 1, and lambda_3 = 3.$

To find the eigenvectors corresponding to each eigenvalue, we need to solve the equation $(A - lambda*I) * v = 0:$

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