Differential Equations | Part IA, 2003

Consider the linear system

x˙(t)Ax(t)=z(t)\dot{\mathbf{x}}(t)-A \mathbf{x}(t)=\mathbf{z}(t)

where the nn-vector z(t)\mathbf{z}(t) and the n×nn \times n matrix AA are given; AA has constant real entries, and has nn distinct eigenvalues λ1,λ2,,λn\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n} and nn linearly independent eigenvectors a1,a2,,an\mathbf{a}_{1}, \mathbf{a}_{2}, \ldots, \mathbf{a}_{n}. Find the complementary function. Given a particular integral xp(t)\mathbf{x}_{\mathbf{p}}(t), write down the general solution. In the case n=2n=2 show that the complementary function is purely oscillatory, with no growth or decay, if and only if

traceA=0 and detA>0.\operatorname{trace} A=0 \quad \text { and } \quad \operatorname{det} A>0 .

Consider the same case n=2n=2 with trace A=0A=0 and detA>0\operatorname{det} A>0 and with

z(t)=a1exp(iω1t)+a2exp(iω2t)\mathbf{z}(t)=\mathbf{a}_{1} \exp \left(i \omega_{1} t\right)+\mathbf{a}_{2} \exp \left(i \omega_{2} t\right)

where ω1,ω2\omega_{1}, \omega_{2} are given real constants. Find a particular integral when

(i) iω1λ1i \omega_{1} \neq \lambda_{1} and iω2λ2i \omega_{2} \neq \lambda_{2};

(ii) iω1λ1i \omega_{1} \neq \lambda_{1} but iω2=λ2i \omega_{2}=\lambda_{2}.

In the case

A=(1251)A=\left(\begin{array}{cc} 1 & 2 \\ -5 & -1 \end{array}\right)

with z(t)=(23i1)exp(3it)\mathbf{z}(t)=\left(\begin{array}{c}2 \\ 3 i-1\end{array}\right) \exp (3 i t), find the solution subject to the initial condition x=(10)\mathbf{x}=\left(\begin{array}{l}1 \\ 0\end{array}\right) at t=0t=0.

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