DEFINITION 7.1.3 (Order of a Differential Equation) The ORDER of a differential equation
is the order of the highest derivative occurring in the equation.
In Example 7.1, the order of Equations 1, 3, 4, 5 are one, that of Equations 2, 6 and 8 are two and
the Equation 7 has order three.
DEFINITION 7.1.4 (Solution) A function
equation (7.1.1) on if
is called a SOLUTION of the differential
is differentiable (as many times as the order of the equation) on
satisfies the differential equation for all
is a solution of an ODE (7.1.1) on
Sometimes a solution
. That is,
, we also say that
is also called an INTEGRAL.
1. Consider the differential equation
. We see that if we take