7 Assessed Exercise 3

A3.1 Calculate the Laplace transforms of (i) $\cos 2\omega t$ and (ii) $\sin^{2}\omega t$ where $\omega>0$ is a constant.

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A3.2 Consider $(n\times n)$ real matrices $A, S, K, L$ . Let $K$ be a positive definite and real symmetric matrix.

(i) Show that if $\lambda$ is an eigenvalue of $K$ , then $\lambda>0$ .

(ii) Deduce that $\det K>0$ and ${\hbox{trace}}(K)>0$ .

(iii) Let $S$ an invertible matrix. Show that $S^{\dagger}KS$ is also positive definite.

(iv) Deduce that $\exp(A^{\dagger})K\exp(A)$ is also positive definite.

(v) Suppose that $L$ is positive definite. Show that $K+L$ is also positive definite.

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A3.3 Solve the initial value problem

{{{dy}\over{dt}}-7y=\sin 2t}

y(0)=0

by taking Laplace transforms. Use partial fractions at the final step of the calculation.

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A3.4 (i) Find the eigenvectors $V$ and eigenvalues of

A=\left[\begin{pmatrix}2&-1&-1\cr-1&2&-1\cr-1&-1&2\cr\end{pmatrix}\right].

(ii) By considering expressions of the form $Y(t)=e^{tw}V,$ find the general solution to

{{d^{2}Y}\over{dt^{2}}}=AY.

This is a model for three identical particles on a common circular track, connected by elastic springs.

(iii) State how many independent constants your solution involves, and explain why this is the correct number.

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