The Handy Math Answer Book

What is a linear differential equation?

A linear differential equation is a first-order differential equation that has no multipli-
cations among the dependent variables and their derivatives, which means the coeffi-
cients are functions of independent variables. Other terms associated with such differ-
ential equations are nonlinear differential equations,which have multiplications
among the dependent variables and their derivatives, and quasi-linear differential
equations,in which a nonlinear differential equation has no multiplications among all
the dependent variables and their derivatives in the highest derivative term.

What are the “solutions”relative to differential equations?

There are three types of “solutions” when discussing differential equations:

General solution—This includes the solutions obtained from integration of the
differential equations. In particular, the general solution of an nth order ordinary dif-
ferential equation has narbitrary constants from integrating ntimes.

Singular solution—Solutions that can’t be expressed by the general solutions.
Particular solution—Solutions obtained from giving specific values to the arbi-
trary constants in the general solution.

What are the “conditions”in reference to differential equations?

In general, there are two “conditions” when discussing differential equations. The first
are called initial conditions,in which constraints are specified at the initial point

dx (,)

dy

=fxy

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MATHEMATICAL ANALYSIS

What is an example of a differential equation?

A

n example of a differential equation involves letting ybe some function of

the independent variable t. Then a differential equation relating yto one or

more of its derivatives is as follows:

In this equation, the first derivative of the function yis equal to the product

of t^2 and the function yitself. This also implies that the stated relationship holds

only for all tfor which both the function and its first derivative are defined.

t

y(t)=t^2 y(t)

u

u