Answer :
Final answer:
To solve the given initial value problem, we need to find the solution to the differential equation x'(t) = [5/7 7/5]x(t) with the initial condition x(0) = [5 1]. We can find the eigenvalues and eigenvectors of the matrix [5/7 7/5], which will allow us to express the general solution. Substituting the initial condition, we can find the specific solution.
Explanation:
To solve the given initial value problem, we need to find the solution to the differential equation x'(t) = [5/7 7/5]x(t) with the initial condition x(0) = [5 1].
We can start by finding the eigenvalues and eigenvectors of the matrix [5/7 7/5]. The eigenvalues are λ = 12 and λ = -2/3, and the corresponding eigenvectors are v₁ = [1 1] and v₂ = [1 -1] respectively.
Using the eigenvalues and eigenvectors, we can express the general solution as x(t) = c₁e^(12t)[1 1] + c₂e^(-2/3t)[1 -1]. Substituting the initial condition x(0) = [5 1], we can find the values of c₁ and c₂, resulting in the specific solution x(t) = [(5/2)e^(12t) + (3/2)e^(-2/3t) (-5/2)e^(12t) + (1/2)e^(-2/3t)].
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