General Solution of First Order Differential Equation

This section contains a brief preview of differential equations. They are studied in more detail in Chapter 14.

A first order differential equation is an equation that involves x, y, and dy/dx. If d²y/dx² also appears in the equation it is called a second order differential equation. The simplest differential equation is

(1)

dy/dx = f(x)

where the function f is continuous on an open interval I.

To solve such an equation we must find a function y = F(x) such that dy/dx = f(x). Differential Equation 1 arises from problems such as the following. Given the velocity v = dy/dt at each time t, find the position y as a function of t. Given the slope dy/dx of a curve at each x, find the curve.

Any antiderivative y = F(x) of f(x) is a solution of this differential equation. Remember that all the antiderivatives of f(x) form a family of functions which differ from each other by a constant.

This family is just the indefinite integral of f,

(1)

The family of functions (Equation 1') is the general solution of the Differential Equation 1.

In this chapter we have solved the problem of finding a nonzero function which is equal to its own derivative. This problem may be set up as another differential equation,

(2)

dy/dx = y.

We found one solution, namely y = e^x. Are there any other solutions?

PROOF

Assume y is a differentiate function of x. The following are equivalent, where x is the independent variable.

In the last step, C = e^C[ if y is positive and C = - e^c' if y is negative.

It can be shown in a similar way that the general solution of the differential equation

(3)

dy/dx = ky,

where k is constant, is

(3')

y = Ce^kx.

The constant C is just the value of y at x = 0,

Ce^k·0 = C.

In applications we often find a differential equation (3) plus an initial condition which gives the value of y at x = 0. The problem can be solved by writing down the general solution of the differential equation and then putting in the value of C given by the initial condition.