Derivatives & Differential Equations
A differential
equation is an equation which contains the derivatives
of a variable, such as the equation
Here x is the variable and the
derivatives are with respect to a second variable t. The letters a, b, c and d
are taken to be constants here. This equation would be described as a second order, linear
differential equation with constant coefficients. It is second order because of
the highest order derivative present, linear because none of the derivatives are
raised to a power, and the multipliers of the derivatives are constant. If x
were the position of an object and t the time, then the first derivative is the
velocity,
the second the acceleration,
and this would be an equation describing the motion of the object. As shown,
this is also said to be a non-homogeneous
equation, and in solving physical problems, one must also consider the homogeneous
equation.