1550078481-Ordinary_Differential_Equations__Roberts_

18 Ordinary Differential Equations

Integrating the second-order differential equation y" = 2x + 1 twice, we find

y' = x^2 + x + c1 and

(2)

Hence, (2) is a two-parameter family of solutions of the DE y" = 2x + 1.

When specifying the number of parameters in a family of functions, we

must be careful to avoid calling every constant a parameter. For example,

the family of functions y = c 1 ex+c^2 contains two constants c 1 and c2, but this

family is not a two-parameter family, since

That is, y = c 1 ex+c^2 is actually the one-parameter family of functions y = kex

disguised as a two-parameter family. When a set of constants { c1, c2, ... , Cn}

in a family of functions cannot be reduced to a small er number by algebraic

manipulation, then the constants are called essential parameters. Hence-

forth, when we say a family of functions is an n-parameter family, we will

assume that it has n essential parameters.

EXAMPLE 1 Verification of a Two-Parameter Family of Solutions

Verify that

(3)

is a two-parameter family of solutions of the second-order, linear differential

equation

(4) y" - y = o.

SOLUTION

Differentiating (3) twice, we get y' = C1 ex - c2e-x and y" = c 1 ex + c2e-x.

Substituting these expressions for y" and y into ( 4), we find

for all x and all choices of c 1 and c 2. Hence, (3) is a solution of the DE (4)
on the interval (-oo, oo) for all choices of c 1 and c2.