ABSTRACT
In studying natural phenomena, we are often interested in more than one quantity and the several quantities may well be connected by differential equations. We are therefore led to consider what happens when more than one differential equation has to be solved at a time. Suppose
a1x˙+ b1y˙ + c1x+ d1y = f1(t), (3.1.1)
a2x˙+ b2y˙ + c2x+ d2y = f2(t) (3.1.2)
where a1, b1, c1, d1, a2, b2, c2, d2 are constants and x, y are to be found. In other words, two simultaneous differential equations of the first order have to be solved. Multiply (3.1.1) by b2 and (3.1.2) by b1. Then subtraction gives
(a1b2 − a2b1)x˙+ αx+ βy = F (t) (3.1.3)
where α = c1b2 − c2b1, β = d1b2 − d2b1 and F (t) = b2f1(t)− b1f2(t). There are two distinct cases to discuss according as a1b2 − a2b1 is or is not
zero. We call a1b2 − a2b1 the test determinant.
