Paper 2, Section II, B

Differential Equations | Part IA, 2015

Suppose that x(t)R3\mathbf{x}(t) \in \mathbb{R}^{3} obeys the differential equation

dxdt=Mx\frac{d \mathbf{x}}{d t}=M \mathbf{x}

where MM is a constant 3×33 \times 3 real matrix.

(i) Suppose that MM has distinct eigenvalues λ1,λ2,λ3\lambda_{1}, \lambda_{2}, \lambda_{3} with corresponding eigenvectors e1,e2,e3\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}. Explain why x\mathbf{x} may be expressed in the form i=13ai(t)ei\sum_{i=1}^{3} a_{i}(t) \mathbf{e}_{i} and deduce by substitution that the general solution of ()(*) is

x=i=13Aieλitei\mathbf{x}=\sum_{i=1}^{3} A_{i} e^{\lambda_{i} t} \mathbf{e}_{i}

where A1,A2,A3A_{1}, A_{2}, A_{3} are constants.

(ii) What is the general solution of ()(*) if λ2=λ3λ1\lambda_{2}=\lambda_{3} \neq \lambda_{1}, but there are still three linearly independent eigenvectors?

(iii) Suppose again that λ2=λ3λ1\lambda_{2}=\lambda_{3} \neq \lambda_{1}, but now there are only two linearly independent eigenvectors: e1\mathbf{e}_{1} corresponding to λ1\lambda_{1} and e2\mathbf{e}_{2} corresponding to λ2\lambda_{2}. Suppose that a vector v\mathbf{v} satisfying the equation (Mλ2I)v=e2\left(M-\lambda_{2} I\right) \mathbf{v}=\mathbf{e}_{2} exists, where II denotes the identity matrix. Show that v\mathbf{v} is linearly independent of e1\mathbf{e}_{1} and e2\mathbf{e}_{2}, and hence or otherwise find the general solution of ()(*).

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