Question

Let \( A \) be a \( 2 \times 2 \) non-diagonalizable real matrix with a real eigenvalue \( \lambda \) and \( v \) be an eigenvector of \( A \) corresponding to \( \lambda \). Which one of the following is the general solution of the system \( y' = Ay \) of first-order linear differential equations?

• A. \( c_1 e^{\lambda t} v + c_2 t e^{\lambda t} v \), where \( c_1, c_2 \in {R} \)
• B. \( c_1 e^{\lambda t} v + c_2 t^2 e^{\lambda t} v \), where \( c_1, c_2 \in {R} \)
• C. \( c_1 e^{\lambda t} v + c_2 e^{\lambda t} (t v + u) \), where \( c_1, c_2 \in {R} \) and \( u \) is a \( 2 \times 1 \) real column vector such that \( (A - \lambda I) u = v \)
• D. \( c_1 e^{\lambda t} v + c_2 e^{\lambda t} (v + u) \), where \( c_1, c_2 \in {R} \) and \( u \) is a \( 2 \times 1 \) real column vector such that \( (A - \lambda I) u = v \)

Hint:

For non-diagonalizable matrices, use generalized eigenvectors to construct solutions of differential equations.

Answer 1

The Correct Option is C

Step 1: General solution for non-diagonalizable matrices. The solution involves a generalized eigenvector \( u \), satisfying \( (A - \lambda I) u = v \). Step 2: Constructing the solution. The solution is: \[ y(t) = c_1 e^{\lambda t} v + c_2 e^{\lambda t} (t v + u). \] Step 3: Conclusion. The correct answer is \( {(2)} \).