MATH 263 prep sheet

y n 1 0 solve for y s l y x find a function tables y

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Unformatted text preview: endent if and only if W [y1( x ), y 2 ( x )] ≠ 0 . 3. • • • f ( t ) g( x − t ) dt L{ f ( x )} ⋅ L{g( x )} an ( x ) dn y d n −1 y dy + an −1 ( x ) n −1 + L + a1 ( x ) + a0 ( x ) = F ( x ) , n dx dx dx Apply Laplace transform to both sides of ODE, using initial conditions to determine y (0), y ′(0),..., y ( n −1) (0) . Solve for Y ( s) = L{ y ( x )} . Find a function tables. y ( x ) with Laplace transform Y ( s) using Eigenvalue of a matrix A: any scalar λ such that Ax = λx for some nonzero vector x Eigenvector of a matrix A: any nonzero vector x such that Ax = λx for some scalar λ Characteristic polynomial of a matrix A: • F ( s) = L{ f ( x )} = ∞ ∫e (λ − λ2 ) L (λ − λk ) Characteristic equation of a matrix A: det (λI − A) = 0 • − sx p(λ ) = det (λI − A) = (λ − λ1 ) Algebraic multiplicity: in the characteristic polynomial, m1 The Laplace Transform c1 f1 ( x ) + c 2 f 2 ( x ) ∫ 1 Eigenvalues If y ' ' = f ( x, y ' ) , let v = y ' , so v ' = y ' ' . dv dy dv . If y ' ' = f ( y , y ' ) , let v = y ' , y '' = ⋅ = ⋅v dy dt dy The equation for v is linear. Solve for v . Integrate v = y ' to find y . f ( x) x e− csF ( s) Solving ODE by Laplace Transforms 1. Reduction of Order 1. e− cs s uc ( x ) f ( x − c ) δ( x ) Given y p ( x) . 4. 5. multiplicity of f ( x ) dx 0 L{c1 f1 ( x )} + L{c 2 f 2 ( x )} F ( s − a) dn [F ( s)] ds n x f ( x) (−1) n dn f dx n s n F ( s) − s n −1 f (0) − s n − 2 f '(0) − ... − f ( n −1) (0) n ⎧0 if x < c uc ( x ) = ⎨ ⎩1 if c < x 0 v1( x),K,v n ( x) . e ax f ( x ) sin(ax ) Look for a particular solution 3. 1 − e−ωs w ww.prep101.com 1 Variation of Parameter...
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