8.3 Assessed exercises 2

Solutions in tutor’s pigeonhole by 17:00 on Tuesday 29th November, please. Feedback is available from the moodle website.

A2.1. (i) Show that the Laplace transform of $f(x)=x$ is $\frac{1}{s^{2}}$.

(ii) Show by induction that the Laplace transform of $\frac{x^{n}}{n!}$ is $\frac{1}{s^{n+1}}$.

(You may assume that $R^{n+1}e^{-sR}\rightarrow 0$ as $R\rightarrow\infty$ for $s>0$.)

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A2.2. Let $g(x,y)={{1}\over{2}}\log(x^{2}+y^{2}).$ By calculating the first and second order partial derivatives, verify that

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A2.3 Three variables $x,t$ and $Y$ are related by the equation

Express $\frac{\partial Y}{\partial t}$, the rate of change of $Y$ with respect to $t$ while keeping $x$ fixed, in terms of $x$ and $Y$.

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