Flashcards
Chapter 2: Second Order ODEs
Cards in this deck (14)
- 1Dirichlet Boundary ConditionsBoundary conditions of the form
y(x0) = y0 - 2Neumann boundary conditionsBoundary conditions of the form
y'(x0) = y0' - 3Robin boundary conditionsBoundary conditions of the form
ay(x0) + by'(x0) = c
Linear combination of Dirichlet and Neumann boundary conditions - 4Wronskiandet([y1 y2; y1' y2'])
W != 0 is a sign of linear independence - 5Superposition principleFor 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 - 6Abel's FormulaW(x) = C*exp(-\int p(x) dx)
- 7Liouville Theoremy2 = y1*\int(W/y1^2)
- 8Homogeneous Solution: Two distinct real rootsy(x) = c1e^(r1*x) + c2e^(r2*x)
- 9Homogeneous Solution: Double real rooty = (c1 + c2x)e^(rx)
- 10Homogeneous Solution: Two distinct complex rootsr = a +- iw
y(x) = e-(a/2)x[Acos(wx) + Bsin(wx)] - 11Euler-Cauchy equation2nd order ODE of the form
x^2*y'' + axy' + by = 0 - 12Solution of Euler-Cauchy equationsTry 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 - 13Equation for Method of Undetermined Coefficientsr(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) - 14Method of Variation of ParametersAssume "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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- Flashcards on Chapter 2: Second Order ODEs, created by Sam Wilson on 08/12/2019.
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- Sam Wilson
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