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Algebra, analysis and differential equations II course of the University of Oxford

Past exam of the algebra, analysis and differential equations II course of the University of Oxford

2011

Differential Equations

The <Laplace transform> is
$$
\overline y(p)=\int_0^\infty e^{-pt}y(t)\,dt.
$$
A sufficient condition is that $y$ be piecewise continuous on each bounded interval and of exponential order $c$, meaning $|y(t)|\leq Ce^{ct}$ for all sufficiently large $t$. Then the integral is absolutely convergent for $\operatorname{Re}p>c$, by comparison with $e^{-(\operatorname{Re}p-c)t}$.

For the constant function, direct integration gives
$$
\boxed{\mathcal L[1](p)=\frac1p\quad(\operatorname{Re}p>0).}
$$

Solution

Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2011 / Differential Equations / 9 / i / Solution

Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2011 / Differential Equations / 9 / i / Solution

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Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2011 / Differential Equations / 9 / i / Solution

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