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Inhomogeneous linear system of differential equation

Find the solution to the given system that satisfies the initial condition

x'(t)= [0,2;4,-2]x(t) + [4t;-4t-2]

a) x(0)= [4;-5]
b) x(2)= [1;1]

Solution Preview

Start from solving the homogeneous system:
x'(t)= [0,2;4,-2]x(t)
To solve is we find the eigenvalues: e_1 = -4 and e_2 = 2, and respective eigenvectors,
v_1 = [1;-2] and v_2 = [1;1] of the matrix [0,2;4,-2].

{ In case you need help in finding them: to find the eigenvalues solve the equation
det( [ 0-e, 2 ; 4, -2-e] ) = 0, and than find any vectors of arbitrary normalization which satisfy equations [ 0-e, 2 ; 4, -2-e] v = 0 }

Now, the ...

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