[PDF] Appendix A - Solutions of Linear Differential Equations





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Appendix A - Solutions of Linear Differential Equations

A.5 Particular Solutions of Linear D.E. with and the Kuhn-Tucker conditions of mathematical progranmiing. ... —rr = fx^x + fu^'^i 6x{0) specified.



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manifold is the de Sitter space Sn+1 with constant curvature one and im Since M is orientable there exists on it a non-zero (n + 1)-form dM (called.

Appendix A

Solution

s o f Linea r

Differentia

l

Equation

s A. l Linea r

Differentia

l

Equation

s wit h

Constan

t

Coefficient

s Linea r diflFerentia l equation s wit h constan t coefficient s ar e usuall y writ ten as 2/(" ai2/("-i a"_i2/(i anV g, (A.l wher e a^ fc = 1,... n ar e numbers y^^^ an d g g{t) i s a know n function of t. We shall denote hy D = ^ the derivative operator^ so that th e differentia l equatio n no w become s p{D)y (D aiD^- i a^_i D an)y g. (A.2 ) If g(t) = 0, the equation is said to be homogeneous. If g{t) ^ 0, then the homogeneou s o r reduce d equatio n i s obtaine d fro m (A.2 b y replacin g g byO. I f y an d y ar e tw o differen t solution s o f (A.2) the n i t i s eas y t o show that y - y* solves the reduced equation of (A.2). Hence, if y is any solutio n t o (A.2) i t ca n b e writte n a s y = y*+y\ (A.3) wher e y* i s an y othe r particula r solutio n t o (A.2 an d y^ i s a suitabl e solutio n t o th e homogeneou s equation

Therefore

solvin g (A.2 involve s (a) finding all the solutions to the homogeneous equation, caUed the gen eral solution, an d (b findin g a particula r solutio n t o th e give n equation

364 A. Solutions of Linear Differential Equations

Th e res t o f thes e note s indicat e ho w t o solv e thes e tw o problems Give n (A.l th e auxiliary equation i s p{m) mP aim^'^ an-im an

0, (A.4)

I n othe r words p{m) i s obtaine d fro m p{D) b y replacin g D b y m Th e auxiliar y equatio n i s a n ordinar y polynomia l o f nt h degre e an d ha s n rea l o r comple x roots countin g multipl e root s accordin g t o thei r multiplicity W e wil l se e that give n thes e roots w e ca n writ e th e genera l solutio n form s o f homogeneou s Unea r differentia l equations A. 2

Homogeneou

s

Equation

s o f Orde r On e Her e th e equatio n i s (D a)y y'-ay 0, whic h ha s y Ce^^ a s it s genera l solutio n form A. 3

Homogeneou

s

Equation

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