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Flashcards by Sam Wilson, updated more than 1 year ago
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Created by Sam Wilson
almost 5 years ago
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Copied by Sam Wilson
almost 5 years ago
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| Question | Answer |
| 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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