Problem 1.

Using LaPlace transforms, solve the equation:

Problem 2.

Solve the following simultaneous differential equations by writing them in standard form and then determining the solution:

Problem 3.

Consider the following differential equation:

(a)  For the homogeneous equation (u(t) = 0), write the system in state space representation (viewgraph 14, lecture 3) and find the solution.

(b)  Now assuming that , write the system in “observable canonical form” (viewgraph 16, lecture 3).  Write out the form of the solution as far as you can go (viewgraph 20).

Problem 4.

Consider the same differential equation as in problem 3,

.

Let   . 

Using the LaPlace transform technique, find the solution to the equation (all initial conditions are zero).

Note: The “characteristic equation” is the polynomial in the variable s that you get when you apply the LaPlace transform to the left hand side of the equation (the y-part) when all initial conditions are zero. 

Note: if you get repeated roots (more than one root with exactly the same value) in the characteristic equation, then consider the section titled “repeated roots” at the bottom of page 43 in Schaum’s.  You can also look up “repeated roots” in the index for Luenberger, but the explanation there is a little bit more opaque).