TAILIEUCHUNG - Handbook of mathematics for engineers and scienteists part 77

The areas of similar triangles are proportional to the squares of the corresponding linear parts (such as sides, altitudes, diagonals, etc.). 4G. The line connecting the midpoints of two sides of a triangle is called a midline of the triangle. The midline is parallel to and half as long as the third side (Fig. ). | 500 Ordinary Differential Equations TABLE Perturbation methods of nonlinear mechanics and theoretical physics the third column gives n leading asymptotic terms with respect to the small parameter e . Method name Examples of problems solved by the method Form of the solution sought Additional conditions and remarks Method of scaled parameters 0 t œ One looks for periodic solutions of the equation y t t oy ef y yt see also Paragraph n-1 y t E ekyk z k 0 n-1 t z 1 È k kj k i 7 Unknowns yk and wk yk 1 yk O 1 secular terms are eliminated through selection of the constants xk Method of strained coordinates 0 t œ Cauchy problem yt f t y e y to yo f is of a special form see also the problem in the method of scaled parameters n-1 y t E ekyk z k 0 n-1 t Z ekpk z k 1 Unknowns yk and pk yk 1 yk O 1 pk 1 pk O 1 Averaging method 0 t œ Cauchy problem y t t 0y ef y yt y 0 yo yt 0 yi for more general problems see Paragraph Item 2 y a t cos p t the amplitude a and phase p are governed by the equations dt -70 fs a d -J - ïïu0 fc a Unknowns a and p fs fo sin pF dp fc 27 X cos pFdp F f a cos p -aw0 sin p Krylov-Bogolyubov-Mitropolskii method 0 t œ One looks for periodic solutions of the equation y t t u2oy ef y yt Cauchy problem for this and other equations n-1 y a cos p ekyk a p k 1 a and p are determined by the equations n d E ekAk a k 1 n d 0 E k k a k 1 Unknowns yk Ak k yk are 2- -periodic functions of p the yk are assumed not to contain cos p Method of two-scale expansions 0 t œ Cauchy problem y t t 0y ef y yt y 0 yo yt 0 yi for boundary value problems see Paragraph Item 2 n-1 y E ekyk Ç n where k 0 ç et n t 1 E k k k 2 7 c _ _ i_ __2 dt e ôî .1 e -J n Unknowns yk and wk yk 1 yk O 1 secular terms are eliminated through selection of œk Multiple scales method 0 t œ One looks for periodic solutions of the equation y t t u0y ef y yt Cauchy problem for this and other equations y E ekyk where k 0 yk yk T0 T1 Tn Tk ekt . . n dt dT0 e dT1 e dTn Unknowns yk

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