Introduction 19DEFINITION General Solution and Singular Solution
If every solution on an interval I of the n-th order differential equation
(5)is obtainable from an n-parameter family of functions(6)by an appropriate choice of the parameters c 1 , c 2 , ... , en, then t h e family of
functions (6) is call ed the general solution of the differential equation (5).A solut ion of an n-th order differential equation which cannot be obtained
from an n-parameter family of solutions is called a singular solution.As we sh all see later in t he text, when the coefficient functions of a lin-
ear differential equation satisfy fairly simple conditions on an interval, then
solutions exist on the interval and all solutions a re obtainable from a fam-
ily of functions. With the exception of a few first-order equations, nonlinear
differential equations are difficult or impossible to so lve explicitly. So for all
practical purposes, the term general solut ion is used only in conjunction with
linear differential equations. The general solution of a differential equation
may not h ave a unique representation as an n-parameter family of functions.
That is, there may b e more than one function of the form (6) which is the
general solution of (5). For example, both of the two-parameter families
and Y2 = k1 sinhx + k2 coshx
a re general solutions of the differential equation y" - y = 0. They are simply
two different ways to represent the same solutions , since
sin. h x =^1 - e x -^1 - e -x and
2 2EXAMPLE 2 Singular Solution of a Differential Equation
a. Verify t hat
(7) y(x) =cx+c^2
is a one-parameter family of solutions of the first-order nonli near differ-
ential equation
(8) (y')^2 + xy' - y = 0.