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 corresponding to these give the values of x +[lambda]1y and
x + [lambda]2y, from which x and y can be found as functions of t,
involving two arbitrary constants. If, however, the two roots of the
quadratic equation for [lambda] are equal, that is, if (a - b')^2 +
4a'b = 0, the method described gives only one equation, expressing x +
[lambda]y in terms of t; by means of this equation y can be eliminated
from dx/dt = ax + by + c, leading to an equation of the form dx/dt =
Px + Q + Re^(a + [lambda]a')t, where P, Q, R are constants. The
integration of this gives x, and thence y can be found.
A similar process is applicable when we have three or more dependent
variables whose differential coefficients in regard to the single
independent variables are given as linear functions of the dependent
variables with constant coefficients.