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T. Tsujioka
Advanced Seminar
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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=a2 + 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 y1(x) and y2(x) are found, L(y1) = L(y2) = 0, then the general solution is obtained by linear combination of them: y = c1y1 + c2y2. Observe Ly = L(c1y1 + c2y2) = c1L(y1) + c2L(y2) = 0. The superposition principle holds for linear operators. Note, however, that two distinct solutions y1(x) and y2(x) must be linearly independent2* to exclude the case where another different set of solutions u1(x) and u2(x) might satisfy the equation Lu(x) = 0. That is, c1y1 + c2y2 0 must hold only if the constants c1, c2 are all 0. By use of orthogonal basis expansion, stemmed from Fourier integral, we first find the characteristic equation of this liner ODE by substituting y = x. We have a2 + b + c = 0, and the solutions are classified by D, the discriminant of the quadratic equation. Let the characteristic polynomial be a2 + b + c corresponding to L=a2 + b+ c. The solution of Ly(x) = 0 is found by the solution of = 0 as follows:

Remark 2*

The set of functions (y1, y2, ..., yn) is linearly independent if and only if the following determinant, called the Wronskian, is0.
W(y, y1, ..., yn)







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