Differential Equations Lecture Work Solutions 35

Differential Equations Lecture Work Solutions 35 - 3 1 f(x...

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3. f ( x ) 1 2 a 0 + X n =1 a n cos L x + b n sin L x ± g ( x ) 1 2 A 0 + X n =1 A n cos L x + B n sin L x ± where a n = 1 L Z L L f ( x )cos L xdx b n = 1 L Z L L f ( x )s in L xdx A n = 1 L Z L L g ( x )cos L xdx B n = 1 L Z L L g ( x )s in L xdx For αf ( x )+ βg ( x )wehave 1 2 γ 0 + X n =1 γ n cos L x + δ n sin L x ± and the coefficients are γ 0 = 1 L Z L L ( αf ( x )+ βg ( x )) dx which by linearity of the integral is γ 0 = α 1 L Z L L f ( x ) dx + β 1 L Z L L g ( x ) dx = αa 0 + βA 0 Similarly for γ n and δ n . γ n = 1 L Z L L ( αf ( x )+ βg ( x )) cos L xdx which by linearity of the integral is γ n = α 1 L Z L L f ( x )cos
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