ABSTRACT
We begin with a general case of the nth order differential equation, which is written in the form resolved for the highest derivative of the unknown function y(x):
…= ′ −( , , , , ),( ) ( 1)y f x y y yn n
(2.1)
where …′ −( , , , , )( 1)f x y y y n is a given function. Obviously, a general solution (general integral) of equation (2.1) depends on n arbitrary
constants. For instance, for the simplest case of equation (2.1):
( ), ( )y f xn =
a general solution obtained by consequent integration of the equation n times contains n arbitrary constants as the coefficients in the polynomial of order n−1:
( ) ( ) . 0
y x dx dx f x dx C xi i
When equation (2.1) describes some phenomena, these constants are related to a concrete situation. For instance, consider Newton’s second law for a one-dimensional motion of a body of mass m moving under the action of a force F: .22m Fd xdt = Integrating, we obtain
′ = ∫ +( ) ( / ) 1x t F m dt C , then assuming constant values of F and m gives ( ) /2 .2 1 2x t Ft m C t C= + + This is a general solution which gives the answer to the problem: x(t) depends quadratically on time. This general solution can be presented as ( ) /2 v2 0 0x t at t x= + + , where
/ /2 2a d x dt F m= = is the acceleration and two arbitrary constants are denoted as v0 and x0.
These constants are the location, (0)0x x= , and velocity, = ′(0)0v x , at time t = 0. For the particular values of x(0) and (0)x ′ given in the concrete situation, we have the particular solution of the problem. Most often, the problems given by differential equations demand a concrete particular solution, but general solutions, like in this example, are often necessary and describe general features of the solution.
