Paper 2, Section I, 2C2 \mathrm{C}

Differential Equations | Part IA, 2019

Consider the first order system

dvdtBv=eλtx\frac{d \boldsymbol{v}}{d t}-B \boldsymbol{v}=e^{\lambda t} \boldsymbol{x}

to be solved for v(t)=(v1(t),v2(t),,vn(t))Rn\boldsymbol{v}(t)=\left(v_{1}(t), v_{2}(t), \ldots, v_{n}(t)\right) \in \mathbb{R}^{n}, where the n×nn \times n matrix B,λRB, \lambda \in \mathbb{R} and xRn\boldsymbol{x} \in \mathbb{R}^{n} are all independent of time. Show that if λ\lambda is not an eigenvalue of BB then there is a solution of the form v(t)=eλtu\boldsymbol{v}(t)=e^{\lambda t} \boldsymbol{u}, with u\boldsymbol{u} constant.

For n=2n=2, given

B=(0310)λ=2 and x=(01)B=\left(\begin{array}{ll} 0 & 3 \\ 1 & 0 \end{array}\right) \quad \lambda=2 \quad \text { and } \boldsymbol{x}=\left(\begin{array}{l} 0 \\ 1 \end{array}\right)

find the general solution to (1).

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