cheat - MA241 Official cheat sheet Undetermined...

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Unformatted text preview: MA241 Official cheat sheet Undetermined coefficients To find a particular solution yp to ay (x) + by (x) + cy(x) = f (x) f includes then yp includes k = multiplicity the root p(x) polynomial of degree m xk (A0 + A1 x + + Am xm ) 0 Cex xk Aex p(x)ex , deg(p) = m xk (A0 + A1 x + + Am xm )ex C1 cos x + C2 sin x xk (A cos x + B sin x) i p(x) cos x + q(x) sin x xk (A0 + A1 x + + As xs ) cos x+ i deg(p) = m, deg(q) = n xk (B0 + B1 x + + Bs xs ) sin x s = max(m, n) C1 ex cos x + C2 ex sin x xk ex (A cos x + B sin x) + i Partial fractions To re-express a fraction of two polynomials r(x) . If q does not q(x) have a higher degree than r, divide r by q r(x) p(x) = h(x) + . q(x) q(x) Factor q into linear and irreducible quadratic polynomials q(x) = a0 (x - a1 )m1 . . . (x - ar )mr (x2 + b1 x + c1 )n1 . . . (x2 + bs x + cs )ns . If the linear factor (x - a) appears m times, then you should consider C2 Cm C1 + + + . 2 x - a (x - a) (x - a)m If the irreducible quadratic factor (x2 + bx + c) appears n times, then you should consider E 1 x + F1 E 2 x + F2 E n x + Fn + + + 2 . (x2 + bx + c) (x2 + bx + c)2 (x + bx + c)n 1 ...
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