Dirac delta and impulse response

Unit-mass pulses shrunk to an instant, the sifting property, transfer functions, and reading a system's reaction to any input off its response to one hammer blow.

Notes on Diffy Qs: Differential Equations for Engineers, Dirac delta and impulse response

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5 exercises

Exercise 101

Solve (find the impulse response) $x'' = \delta(t)$ , $x(0) = 0$ , $x'(0)=0$ .

2.5

Exercise 102

Solve (find the impulse response) $x' + a x = \delta(t)$ , $x(0) = 0$ .

2.5

Exercise 103

Suppose that $L x = \delta(t)$ , $x(0) = 0$ , $x'(0) = 0$ , has the solution $x(t) = e^t \sin(t)$ for $t > 0$ . Find (in closed form) the solution to $Lx = e^t$ , $x(0) = 0$ , $x'(0) = 0$ for $t > 0$ .

2.5

Exercise 104

Compute ${\mathcal{L}}^{-1} \left\{ \frac{s^2}{s^2+1} \right\}$ .

2.5

Exercise 105

Compute ${\mathcal{L}}^{-1} \left\{ \frac{3 s^2 e^{-s} + 2}{s^2} \right\}$ .

2.5