Question 1

What is the Laplace transform of $$\displaystyle{x(t) = e^{-t}\sin{\pi t}}$$?

Question 2

The Laplace transform of $$d^3 x(t)/dt^3$$ is given by

Question 3

If $$\displaystyle{\ddot x 5\dot x 6x = e^t}$$, with $$x(0)=\dot x(0)=0$$, what is $$X(s)$$?

Question 4

If $$\displaystyle{X(s) = \frac{1}{(s-1)(s 1)}}$$, what is $$x(t)$$?

Question 5

The above pictured function can be defined using Heaviside step functions as

Question 6

The inverse Laplace transform of the function $$\displaystyle{ X(s) = \frac{\omega e^{-\pi s/\omega}}{s^2 \omega^2}}$$ is given by

Question 7

The solution to $$\ddot x - x = 1-u_1(t)$$, with $$x(0)=\dot x(0)=0$$, is given by

Question 8

The solution to $$\ddot x - x = \delta(t) - \delta(t-1)$$, with $$x(0)=\dot x(0)=0$$, is given by

Question 9

The general solution to $$y'' - 2xy' y = 0$$ is given by

Question 10

The general solution to $$(x^2 1)y'' - 4xy' 6y =0$$ is given by