ORDINARY LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS To solve the equation
with the initial values x0, xlt • • •, xn_1 for x, dxjdt, • • •, dn~1xjdtn’1 when t = 0, where аъ a2, ■ • ■ are constants., we transform every term of the equation into its Laplace transform. Remembering the differentiation rule, we obtain the transformed equation:
examples appear in the classical theory of airplane dynamics. When the deformation of the main structure is described by a few generalized coordinates, ordinary linear differential equations are obtained. The principal assumptions responsible for such great simplification are (1) that the motion of the airplane consists of infinitesimal disturbances about a steady symmetrical flight, and (2) that the dependance of the aerodynamic forces on the rate of change of linear and angular velocities can be neglected except for the lag of the downwash between the tail and the wing. With the second assumption the unsteady aerodynamic action is expressed as a linear function of the linear and angular velocities, the coefficients being defined ‘as “stability derivatives.”
The extensive and intricate subject of airplane dynamics, however, will not be discussed here. (See Refs. 10.10-10.13.) In the following sections, illustrations of the application of Laplace transformation will be given in the gust-response and flutter problems.