Chapter 2: Second Order ODEs

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Karteikarten am Chapter 2: Second Order ODEs, erstellt von Sam Wilson am 08/12/2019.
Sam Wilson
Karteikarten von Sam Wilson, aktualisiert more than 1 year ago
Sam Wilson
Erstellt von Sam Wilson vor fast 5 Jahre
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Dirichlet Boundary Conditions Boundary conditions of the form y(x0) = y0
Neumann boundary conditions Boundary conditions of the form y'(x0) = y0'
Robin boundary conditions Boundary conditions of the form ay(x0) + by'(x0) = c Linear combination of Dirichlet and Neumann boundary conditions
Wronskian det([y1 y2; y1' y2']) W != 0 is a sign of linear independence
Superposition principle For an equation G(x,y,y',y''), a linear combination of any two particular solutions y1 and y2 is also a solution to G() y = ay1 + by2 iff y1 and y2 are linearly independent
Abel's Formula W(x) = C*exp(-\int p(x) dx)
Liouville Theorem y2 = y1*\int(W/y1^2)
Homogeneous Solution: Two distinct real roots y(x) = c1e^(r1*x) + c2e^(r2*x)
Homogeneous Solution: Double real root y = (c1 + c2x)e^(rx)
Homogeneous Solution: Two distinct complex roots r = a +- iw y(x) = e-(a/2)x[Acos(wx) + Bsin(wx)]
Euler-Cauchy equation 2nd order ODE of the form x^2*y'' + axy' + by = 0
Solution of Euler-Cauchy equations Try y = x^m Plug into equation and you'll get a quadratic equation in m Solve for m and use it as the root to find your solution use ln(x) instead of x
Equation for Method of Undetermined Coefficients r(x) = e^(yx)(Pl(x)cos(wx) + Qm(x)sin(wx)) particular solution gets multiplied by x^k and P and Q are represented as polynomial of order min(l,m)
Method of Variation of Parameters Assume "constants" are functions of x Differentiate Use constraint c1'y1 + c2'y2 = 0 Form SLE with constraint and equation Solve Solution involves Wronskian and r(x)
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