dx/dt = ax + by + c, dy/dt = a'x + b'y + c',
consists in differentiating the first equation, thereby obtaining
d^2x dx dy
---- = a-- + b--;
dt^2 dt dx
from the two given equations, by elimination of y, we can express
dy/dt as a linear function of x and dx/dt; we can thus form an
equation of the shape d^2x/dt^2 = P + Qx + Rdx/dt, where P, Q, R are
constants; this can be integrated by methods previously explained, and
the integral, involving two arbitrary constants, gives, by the
equation dx/dt = ax + by + c, the corresponding value of y. Conversely
it should be noticed that any single linear differential equation
d^2x dx
---- = u + vx + w--,
dt^2 dt
where u, v, w are functions of t, by writing y for dx/dt, is
equivalent with the two equations dx/dt = y, dy/dt = u + vx + wy. In
fact a similar reduction is possible for any system of differential
equations with one independent variable.