Page 19 - Schaum's Outline of Differential Equations

2 BASIC CONCEPTS [CHAR 1

NOTATION
The expressions /, /', /", y- ',..., y-"' are often used to represent, respectively, the first, second, third, fourth,
2
..., wth derivatives of y with respect to the independent variable under consideration. Thus, /' represents d yldx 2
2
if the independent variable is x, but represents d yldp 2 if the independent variable is p. Observe that parentheses
n
are used in y^ to distinguish it from the wth power, y^ \ If the independent variable is time, usually denoted by t,
2
2
3
3
primes are often replaced by dots. Thus, y, y, and y represent dyldt, d yldt , and d yldt , respectively.
SOLUTIONS
A solution of a differential equation in the unknown function y and the independent variable x on the interval
J>, is a function y(x) that satisfies the differential equation identically for all x in J>.

Example 1.4. Is y(x) = c 1 sin 2x + c 2 cos 2x, where c 1 and c 2 are arbitrary constants, a solution of y" + 4y = 0?
Differentiating y, we find

Hence,

Thus, v = Ci sin 2jc + c, cos 2x satisfies the differential equation for all values of x and is a solution on the interval (- °°, °°).

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Example 1.5. Determine whether y = x — 1 is a solution of (y') 4 + y 2 = —I.
Note that the left side of the differential equation must be nonnegative for every real function y(x) and any x, since it
is the sum of terms raised to the second and fourth powers, while the right side of the equation is negative. Since no function
y(x) will satisfy this equation, the given differential equation has no solution.

We see that some dinerential equations have mnnitely many solutions (Example 1.4), whereas other dil-
ferential equations have no solutions (Example 1.5). It is also possible that a differential equation has exactly
2
one solution. Consider (y') 4 + y = 0, which for reasons identical to those given in Example 1.5 has only one
solution y = 0.
A particular solution of a differential equation is any one solution. The general solution of a differential
equation is the set of all solutions.

Example 1.6. The general solution to the differential equation in Example 1.4 can be shown to be (see Chapters 8 and 9)
y = Ci sin 2x + c 2 cos 2x. That is, every particular solution of the differential equation has this general form. A few particular
solutions are: (a) y = 5 sin 2x - 3 cos 2x (choose c 1 = 5 and c 2 = — 3), (b) y = sin 2x (choose c 1 = 1 and c 2 = 0), and (c)y = 0
(choose Ci = c, = 0).

The general solution of a differential equation cannot always be expressed by a single formula. As an example
2
consider the differential equation y' + y = 0, which has two particular solutions y = \lx and y = 0.

INITIAL-VALUE AND BOUNDARY-VALUE PROBLEMS
A differential equation along with subsidiary conditions on the unknown function and its derivatives, all
given at the same value of the independent variable, constitutes an initial-value problem. The subsidiary condi-
tions are initial conditions. If the subsidiary conditions are given at more than one value of the independent
variable, the problem is a boundary-value problem and the conditions are boundary conditions.