I, where t is the time.. Equation 3 governs the angular motion Bt 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..
Trang 27dz
- = 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-
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