M 427K - Syllabus - Math 427k ordinary, partial diff....

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Unformatted text preview: Math 427k ordinary, partial diff. eqns.: defs linear d.e.’s order of a d.e. solution, general solution of a d.e. direction field 1st order linear: y + a(x)y + b(x) = 0 µ = e a(x) dx use of definite integrals for init. conds. existence and uniqueness for: 1st order linear general 1st order separable d.e. exact d.e.: M + N dy/dx = 0 My − Nx = 0 for exact φ = M dx + h(y ) φ = C is solution integrating factor µ(x, y ) [My − Nx ]/N = function of x implies µ = µ(x) = e [My −Nx ]/N dx [My − Nx ]/M = function of y implies µ = µ(y ) = e− [My −Nx ]/M dy homogeneous: y = F (y/x) set v = y/x, becomes separable nd 2 order linear existence-uniqueness homogeneous (second meaning) linear operator L fundamental set of sols Wronskian W (f, g ) = f g − gf connection between Wronskian and existence of fundamental set of sols linear independence, and connection with Wronskian reduction of order (homogeneous) constant coeffs particular sols of inhomogeneous d.e. variation of parameters review of sequences and series power series radius of convergence ratio test for radius differentiating and integrating series Taylor series, uniqueness ordinary point sols near ordinary point regular singular point sols near reg sing point boundary value problems L(y ) = λy ay (x1 ) + by (x1 ) = 0 cy (x2 ) + dy (x2 ) = 0 eigenvalues, eigenfunctions partial differential equations separation of variables heat equation Fourier series for interval [−a, a] cosine and sine series for [0, a] variation of above wave equation Laplace equation numerical solution of o.d.e. Euler method Runge-Kutta method ...
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This note was uploaded on 09/18/2011 for the course M 427K taught by Professor Fonken during the Spring '08 term at University of Texas at Austin.

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