Non-Homogeneous Equations y(x) = y c + y p if f(x) ≠ y c y p = A*f(x) (when f(x) is exp or polynomial) y p = Asin(x) + Bcos(x) (when f(x) is sin/cos) polynomial terms: (Ax 2 + Bx + C) even if some of the coefficients are 0if f(x) = y c multiply “bad” term by smallest power of x that will eliminate duplication Variation of Parameters If y’’ + P(x)y’ + Q(x)y = f(x) & y c =c 1 y 1 + c 2 y 2 Y p = -y 1 ∫ [y 2 f(x))/w] + y 2 ∫ [y 1 f(x)]/w Where w=(y 1 )(y’ 2 ) – (y 2 )(y’ 1 ) Undamped Forced Oscillation mx’’ + kx = F0 cos( ϖ t); ϖ0 = √ (k/m) x c = c 1 cos( ϖ0 t) + c 2 sin( ϖ0 t) x p = [(F0 /m)/( ϖ0 2-ϖ0 )]cos( ϖ t) if ϖ 0 = ϖ resonance ϖ0 is internal & ϖ is external frequency Damped Forced Oscillations mx’’ + cx’ + kx = F(t) x c is the transient solution x p = stable solution amplitude of x p is called forced amplitude. It’s max value is called practical resonance Method of Elimination Solve one equation in terms of another Take the derivative of one equation
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This note was uploaded on 10/06/2008 for the course MATH 216 taught by Professor Stenstones? during the Spring '07 term at University of Michigan.