d^2y1
-- = y1------ + [eta]--- and --- = y1------- + 2--- ------ + [eta]-----,
dx dx dx dx^2 dx^2 dx dx dx^2

and thus

d^2y dy d^2[eta] / dy1 \ d[eta] /d^2y1 dy1 \
--- + P -- + Qy = y1------- + ( 2--- + Py1) ------ + ( ----- + P--- + Qy1)[eta];
dx^2 dx dx^2 \ dx / dx \ dx^2 dx /

if then

d^2y1 dy1
---- + P --- + Qy1 = 0,
dx^2 dx

and z denote d[eta]/dx, the original differential equation becomes

dz / dy1 \
y1-- + ( 2--- + Py1)z = R.
dx \ dx /

From this equation z can be found by the rule given above for the
linear equation of the first order, and will involve one arbitrary
constant; thence y = y1 [eta] = y1 [int] zdx + Ay1, where A is another
arbitrary constant, will be the general solution of the original
equation, and, as was to be expected, involves two arbitrary
constants.

The case of most frequent occurrence is that in which the coefficients
P, Q are constants; we consider this case in some detail. If [t]*
be a root of the quadratic equation [t]^2 + [t]P + Q = 0, it
can be at once seen that a particular integral of the differential
equation with zero on the right side is y1 = e^[theta]x. Supposing
first the roots of the quadratic equation to be different, and [phi]
to be the other root, so that [p] + [t] = -P, the auxiliary
differential equation for z, referred to above, becomes dz/dx +
([t] - [p])z = Re^(-[t]^x), which leads to
ze^{([t]-[p])^x} = B + [int] Re^(-[p]^x)dx, where B is an
arbitrary constant, and hence to

(*) [t] = [theta]; [p] = [phi].
_ _ _
/ / /
y = Ae^([t]^x) + e^([t]^x)| Be^([p]-[t])^x dx + e^[t]^x | e^([p]-[t])^x | Re^-[p]^x dxdx,
_/ _/ _/

or say to y = Ae^[t]^x + Ce^[p]^x + U, where A, C are arbitrary
constants and U is a function of x, not present at all when R = 0. If
the quadratic equation [t]^2 + P[t] + Q = 0 has equal roots, so that
2[t] = -P, the auxiliary equation in z becomes dz/dx = Re^-[t]^x,
giving z = B + [int] Re^-[t]^x dx, where B is an arbitrary constant,
and hence
_ _