Mathematical Tools for Physics

4—Differential Equations 101

4.7 Simultaneous Equations
What’s this doing in a chapter on differential equations? Patience. Solve two equations in two unknowns:

(X)ax+by=e

(Y)cx+dy=f

d×(X)−b×(Y):

adx+bdy−bcx−bdy=ed−fb

(ad−bc)x=ed−fb

Similarly, multiply (Y) byaand (X) bycand subtract:

acx+ady−acx−cby=fa−ec

(ad−bc)y=fa−ec

Divide by the factor on the left side and you have

x=

ed−fb

ad−bc

, y=

fa−ec

ad−bc

(26)

provided thatad−bc 6 = 0. This expression appearing in both denominators is the determinant of the equations.
Classify all the essentially different cases that can occur with this simple-looking set of equations and draw
graphs to illustrate them. If this looks like problem1.23, it should.

y

1.

x

y

2.

x

y

3a.

x

1. The solution is just as I found it above and nothing goes wrong. There is one and only one solution.
   The two graphs of the two equations are two intersecting straight lines.


2. The denominator, the determinant, is zero and the numerator isn’t. This is impossible and there are no
   solutions. When the determinant vanishes, the two straight lines are parallel and the fact that the numerator isn’t
   zero implies that the two lines are distinct and never intersect. (This could also happen if in one of the equations,
   say (X),a=b= 0ande 6 = 0. For example0 = 1. This obviously makes no sense.)