Mathematical Tools for Physics

4—Differential Equations 102

3a. The determinant is zero and so are both numerators. In this case the two lines are not only parallel,
they are the same line. The two equations are not really independent and you have an infinite number of solutions.
3b. You can get zero over zero another way. Both equations (X) and (Y) are0 = 0. This sounds trivial,
but it can really happen.Everyxandywill satisfy the equation.

4. Not strictly a different case, but sufficiently important to discuss it separately: Suppose the the right-
   hand sides of (X) and (Y) are zero,e=f= 0. If the determinant is non-zero, there is a unique solution and it
   isx= 0,y= 0.


5. Withe=f= 0, if the determinant is zero, the two equations are the same equation and there are an
   infinite number of non-zero solutions.
   In the important case for whiche=f= 0and the determinant is zero, there are two cases: (3b) and (5).
   In the latter case there is a one-parameter family of solutions and in the former case there is a two-parameter
   family. Put another way, for case (5) the set of all solutions is a straight line, a one-dimensional set. For case
   (3b) the set of all solutions is the whole plane, a two-dimensional set.

y

4.

x

y

5.

x

y

3b.

x

Example: Consider the two equations

kx+ (k−1)y= 0, (1−k)x+ (k−1)^2 y= 0

For whatever reason, I would like to get a non-zero solution forxandy. Can I? The condition depends on the
determinant, so I take the determinant and set it equal to zero.

k(k−1)^2 −(1−k)(k−1) = 0, or (k+ 1)(k−1)^2 = 0

There are two roots,k=− 1 andk= +1. In thek=− 1 case the two equations become

−x− 2 y= 0, and 2 x+ 4y= 0