ta]
------ = A + B[eta] + C[eta]^2,
dx

where A, B, C are functions of x, is, by the substitution

1 dy
[eta] = - -- --,
Cy dx

reduced to the linear equation

d^2y / 1 dC\ dy
---- - ( B + - -- )-- + ACy = 0.
dx^2 \ C dx/ dx

The equation

d[eta]
------ = A + B[eta] + C[eta]^2,
dx

known as Riccati's equation, is transformed into an equation of the
same form by a substitution of the form [eta] = (aY + b)/(cY + d),
where a, b, c, d are any functions of x, and this fact may be utilized
to obtain a solution when A, B, C have special forms; in particular if
any particular solution of the equation be known, say [eta]0, the
substitution [eta] = [eta]0 - 1/Y enables us at once to obtain the
general solution; for instance, when

d /A\
2B = -- log( - ),
dx \C/

a particular solution is [eta]0 = [root](-A/C). This is a case of the
remark, often useful in practice, that the linear equation

d^2y d[phi] dy
[phi](x)---- + 1/2------ -- + [mu]y = 0,
dx^2 dx dx

where [mu] is a constant, is reducible to a standard form by taking a
new independent variable
_
/
z = | dx[[p](x)]^-1/2.
_/

We pass to other types of equations of which the solution can be
obtained by rule. We may have cases in which there are two dependent
variables, x and y, and one independent variable t, the differential
coefficients dx/dt, dy/dt being given as functions of x, y and t. Of
such equations a simple case is expressed by the pair

dx dy
-- = ax + by + c, -- = a'x + b'y + c',
dt dt

wherein the coefficients a, b, c, a', b', c', are constants. To
integrate these, form with the constant [lambda] the differential
coefficient of z = x + [lambda]y, that is dz/dt = (a + [lambda]a')x +
(b + [lambda]b')y + c + [lambda]c', the quantity [lambda] being so
chosen that b + [lambda]b' = [lambda](a + [lambda]a'), so that we have
dz/dt = (a + [lambda]a')z + c + [lambda]c'; this last equation is at
once integrable in the form z(a + [lambda]a') + c + [lambda]c' = Ae^(a
+ [lambda]a')t, where A is an arbitrary constant. In general, the
condition b + [lambda]b' = [lambda](a + [lambda]a') is satisfied by
two different values of [lambda], say [lambda]1, [lambda]2; the
solutions co