Paper 2, Section II, B

(i) Find, in the form of an integral, the solution of the equation

$\alpha \frac{d y}{d t}+y=f(t)$

that satisfies $y \rightarrow 0$ as $t \rightarrow-\infty$. Here $f(t)$ is a general function and $\alpha$ is a positive constant.

Hence find the solution in each of the cases:

(a) $f(t)=\delta(t)$;

(b) $f(t)=H(t)$, where $H(t)$ is the Heaviside step function.

(ii) Find and sketch the solution of the equation

$\frac{d y}{d t}+y=H(t)-H(t-1)$

given that $y(0)=0$ and $y(t)$ is continuous.

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