Paper 2, Section II, B

Differential Equations | Part IA, 2007

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

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

that satisfies y0y \rightarrow 0 as tt \rightarrow-\infty. Here f(t)f(t) is a general function and α\alpha is a positive constant.

Hence find the solution in each of the cases:

(a) f(t)=δ(t)f(t)=\delta(t);

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

(ii) Find and sketch the solution of the equation

dydt+y=H(t)H(t1)\frac{d y}{d t}+y=H(t)-H(t-1)

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

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