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M 427K - Syllabus

# M 427K - Syllabus - Math 427k ordinary partial di eqns defs...

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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 1 st order linear: y 0 + a ( x ) y + b ( x ) = 0 μ = e R a ( x ) dx use of definite integrals for init. conds. existence and uniqueness for: 1 st order linear general 1 st order separable d.e. exact d.e.: M + Ndy/dx = 0 M y - N x = 0 for exact φ = R M dx + h ( y ) φ = C is solution integrating factor μ ( x, y ) [ M y - N x ] /N = function of x implies μ = μ ( x ) = e R [ M y - N x ] /N dx [ M y - N x ] /M = function of y implies μ = μ ( y ) = e - R [ M y - N x ] /M dy homogeneous: y 0 = F ( y/x ) set v = y/x , becomes separable 2 nd order linear existence-uniqueness homogeneous (second meaning) linear operator L fundamental set of sols Wronskian W ( f, g ) = fg 0 - gf 0 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.
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