The Handy Math Answer Book

What are first-order homogeneous and

non-homogeneous linear differential equations?

T

hese differential equations may be long-winded phrases, but they are actually

types of first-order differential equations. A first-order homogeneous linear

differential equation can be written in the notation as follows:

The first-order homogeneous linear differential equations are those that

place all terms that include the unknown equation and its derivative on the left-

hand side of the equation; on the right-hand side, it is set equal to zero for all t.

The non-homogeneous linear differential equations are those that, after iso-

lating the linear terms containing y(t) and the partial differentials inside the

above large parentheses on the left side of the equation, do not set the right-

hand side identically to zero. It is often represented by one function, such as the

b(t) (see below). The standard notation is as follows:

(usually in reference to time); such problems are called initial value problems. The

other condition is the boundary condition,in which constraints are specified this time

at the boundary points (usually in reference to space), with such problems called

boundary value problems.

What techniques are used to solve first-order differential equations?

There are usually three major ways to solve first-order differential equations: analyti-

cally, qualitatively, and numerically. The analytical way includes the examples men-

tioned above, such as the linear and separable equations. Qualitative methods include

such methods as defining the slope of a field of a differential equation. Finally, numer-

ical techniques can be thought of as something close to Euler’s method, a way of find-

ing the largest divisor of two numbers.

What is Euler’s method?
Swiss mathematician Leonhard Euler (1707–1783) was one of the most prolific math-
ematicians who ever lived. He developed Euler’s method, which is a way of determin-
234 ing the largest divisor of two numbers. For example, if we want to find the largest divi-

(ty(t))+ a(t) y(t)=0

(uty(t))+ a(t) y(t)=b(t)

u

u

u