dz - = C Z , dt

d211 d ( 2 1 2 - = C 1 + - , dx"

Figure 1, Electrical circuit,

(6) equation (2).

Equation (1) is the differential equatio11 governing the linear displacement z ( t ) of a body of mass m, subjected to an applied force F(t ) and a restraining spring of stiffness k , as mentio~led in the preceding section.

Equation (2) governs the current i ( t ) in an electrical circuit containing an in- ductor with inductance L, a capacitor with capacitance C, and an applied voltage source of strength E( t ) (Fig. I), where t is the time.

Equation (3) governs the angular motion B(t) of a pe~ldulum of length 1, u~lder the action of gravity, where g is the acceleration of gravity and t is the time (Fig. 2).

Equation (4) governs the population .c(t) of a single species, where t is the time and c is a net birthldeath rate constant.

Equation (5) governs the shape of a flexible cable or string, hanging under the action of gravity, where y(z) is the deflection and C is a constant that depends upon the mass density of the cable and the tension at the inidpoint z = 0 (Fig. 3). Figure 2. Pendulum, equation (3).

Finally, equation (6) governs the deflection y(z) of a beam subjected to a load- ing zu(x) (Fig. 4), where E and I are physical constants of the beam material and cross section, respectively.

Ordinary and partial differential equations. We classify a differential equa- tion as an ordinary differential equation if it contains ordinary derivatives with respect to a single independent variable, and as a partial differential equation if x it contains partial derivatives with respect to two or more independent variables.

Figure 3. Hanging cable, Thus, equations ( I ) - (6) are ordinary differential equations (often abbreviated as ODE'S). The independent variable is t in (1)-(4) and x in (5) and (6). equatlon (5).

Some representative and i~nportant partial differential equations (PDE's) are as follows:

a 2 u 821 C U 2 = -

dz-t ' (7)

Figure 4. Loaded beam,

(lo) equation (6).