Title:

Author:

Mechanics

Copyright:

T. Tsujioka

Advanced Seminar

Mechanics

0. Mathematical Preliminaries

2. Differential Equations

2-2. 2nd order Linear ODEs (Ordinary Differential Equations)

__Consider the 2nd order linear differential equations of this type:

____*ay*'' + *by*' + *cy* = *r*(*x*) ---------------------------------------------... (Eq. 2-1)

which may be encountered in situations of many mechanical systems. We here use a convenient operator method.

Let *L* be a differential linear operator defined by

____.

Or, if we put *d*/*dx*, *L*=*a*^{2} + *b*+ *c*. Then, (Eq. 2-1) can be written as *Ly*(*x*) = *r*(*x*).

__Case I: *r*(*x*) = 0: *Ly*(*x*) = 0. The equation is said to be *homogeneous*. If two distinct solutions *y*_{1}(*x*) and *y*_{2}(*x*) are found, *L*(*y*_{1}) = *L*(*y*_{2}) = 0, then the general solution is obtained by linear combination of them: *y* = *c*_{1}*y*_{1} + *c*_{2}*y*_{2}. Observe *Ly* = *L*(*c*_{1}*y*_{1} + *c*_{2}*y*_{2}) = *c*_{1}*L*(*y*_{1}) + *c*_{2}*L*(*y*_{2}) = 0. The superposition principle holds for linear operators. Note, however, that two distinct solutions *y*_{1}(*x*) and *y*_{2}(*x*) must be __linearly independent ^{2*}__ to exclude the case where another different set of solutions

__Remark__ 2*

The set of functions (*y*_{1}, *y*_{2}, ..., *y _{n}*) is linearly independent if and only if the following determinant, called the Wronskian, is0.

W(