Order the steps for solving a 2×2 linear system X' = AX.

Write the system in matrix form X' = AX → Find eigenvalues of A: det(A - λ I) = 0 → Find eigenvectors for each eigenvalue → Write general solution X = c1v1e^λ1t + c2v2e^λ2t → Apply initial conditions to find c1 and c2

Physics I / Systems of Differential Equations

Practice question · Put in order

Order the steps for solving a $2\times2$ linear system $X' = AX$ .

Write the system in matrix form X' = AX
Find eigenvectors for each eigenvalue
Apply initial conditions to find $c_{1}$ and $c_{2}$
Write general solution $X = c_{1}v_{1}e^{\lambda_{1}t} + c_{2}v_{2}e^{\lambda_{2}t}$
Find eigenvalues of A: $det(A - \lambda I) = 0$

Hints

The eigenvalues of the coefficient matrix come before anything else.
Each eigenvector supplies one basic solution to combine.

Show the answer

Write the system in matrix form X' = AX
Find eigenvalues of A: $det(A - \lambda I) = 0$
Find eigenvectors for each eigenvalue
Write general solution $X = c_{1}v_{1}e^{\lambda_{1}t} + c_{2}v_{2}e^{\lambda_{2}t}$
Apply initial conditions to find $c_{1}$ and $c_{2}$

Why

Eigenvalues and eigenvectors of A determine the system's solution.

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Order the steps for solving a $2\times2$ linear system $X' = AX$ .