Differential Equations | Part IA, 2006

Consider the first order system

dxdtAx=eλtv\frac{d \mathbf{x}}{d t}-A \mathbf{x}=e^{\lambda t} \mathbf{v}

to be solved for x(t)=(x1(t),x2(t),,xn(t))Rn\mathbf{x}(t)=\left(x_{1}(t), x_{2}(t), \ldots, x_{n}(t)\right) \in \mathbb{R}^{n}, where AA is an n×nn \times n matrix, λR\lambda \in \mathbb{R} and vRn\mathbf{v} \in \mathbb{R}^{n}. Show that if λ\lambda is not an eigenvalue of AA there is a solution of the form x(t)=eλtu\mathbf{x}(t)=e^{\lambda t} \mathbf{u}. For n=2n=2, given

A=(0100),λ=1, and v=(11)A=\left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right), \quad \lambda=1, \quad \text { and } \quad \mathbf{v}=\left(\begin{array}{l} 1 \\ 1 \end{array}\right)

find this solution.

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