MATH 263 prep sheet

A0 a1k an and b0 b1k bm yp into weve helped over

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Unformatted text preview: ) takes the form: An x n + An −1 x n −1 + L + A1 x + A0 eαx (An x n + An −1 x n −1 + L + A1 x + A0 ) eαx cos(βx )(An x n + An −1 x n −1 + L + A1 x + A0 ) +eαx sin(βx )(Bm x m + Bm −1 x m −1 + L + B1 x + B0 ) ay ′′ + by ′ + cy = f ( x ) . A0, A1,K, An and B0, B1,K, Bm . yp into We’ve helped over 50,000 students get better grades since 1999! Need help for exams? Check out our classroom prep sessions - customized to your exact course - at www.prep101.com Cauchy-Euler Equation: x 2 y '' + axy ' + by = f ( x ), a, b constant The homogeneous solution is determined by the roots of the auxiliary equation: λ2 + (a − 1)λ + b = 0 . x ∫ F ( s) s f ( t ) dt 0 ω f ( x) Roots Fundamental Solutions {x λ , x λ } C1x λ1 + C2 x λ2 equal {x λ , x λ log x} C1 x λ + C2 x λ log x x x [C1 cos(β log x ) + C2 sin(β log x )] ω e− ax ∫e 0 Homogeneous Solution distinct periodic with period λ1 ≠ λ2 1 λ1 = λ2 λ = α ± iβ {x complex α 2 cos(β log x ), x sin(β log x )} α α 1. Find the general solution y h ( x ) = c1 y1 ( x ) + L + c n y n ( x ) to the homogeneous equation dn y d n −1 y dy an ( x ) n + an −1 ( x ) n −1 + L + a1 ( x ) + a0 ( x ) = 0 . dx dx dx 2. s2 1 s+ a s s2 + a 2 a s2 + a 2 Plug y p ( x ) = v1 ( x ) y1 ( x ) + L + v n ( x ) y n ( x ) . y p ( x ) into the original equation and solve for v1( x ),K, v n ( x ) into Plug The final answer is y ( x ) = y h ( x ) + y p ( x ) . Wronskian: W [y1 ( x ), y 2 ( x )] = • f ( x ) * g( x ) = 2. 3. 4. 2. y1 ( x ) y 2 '( x ) − y 2 ( x ) y1 '( x ) y1( x ), y 2 ( x ) are indep...
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