FODE_11-15

FODE_11-15 - Tutorial FODE Ordinary First-Order...

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Tutorial FODE: Ordinary First-Order Differential Equations yields y ( x )= Ce ¡ 3 x + x ¡ 1 3 : The condition gives 3 = C ¡ 1 = 3 = ) C = 10 = 3 ; or y ( x ) = 1 3 ¡ 10 e ¡ 3 x +3 x ¡ 1 ¢ : b. Take y p = Ae x : Then, Ae x ¡ 2 Ae x = e x = ) ¡ A = 1 or A = ¡ 1 : The homogenous solution is y 0 = Ce 2 x : Thus, y ( x )= Ce 2 x ¡ e x : The condition gives 1 = C ¡ 1 = ) C =2 ; or y ( x )= 2 e 2 x ¡ e x : c. Take y p = Ax 3 + Bx 2 + Cx + D: Then 3 Ax 2 +2 Bx + C +2 ¡ Ax 3 + Bx 2 + Cx + D ¢ = x 3 ; or 2 A = 1 3 A +2 B = 0 2 B +2 C = 0 C +2 D = 0 from which A =1 = 2 ; B = ¡ 3 = 4 ; C =3 = 4 and D = ¡ 3 = 8 : The homogeneous solution is y 0 = Ee ¡ 2 x : Hence, y ( x )= Ee ¡ 2 x + 1 8 ¡ 4 x 3 ¡ 6 x 2 +6 x ¡ 3 ¢ : The condition gives 2 = Ee ¡ 2 + (4 ¡ 6 +6 ¡ 3) = 8 = ) E =15 e 2 = 8 Thus, y ( x ) = 15 8 e 2(1 ¡ x ) + 1 8 ¡ 4 x 3 ¡ 6 x 2 +6 x ¡ 3 ¢ : d. Take y p = A cos x + B sin x: Then ¡ A sin x + B cos x ¡ A cos x ¡ B sin x = 2cos x = ) ¡ A ¡ B = 0 ; B ¡ A = 2 =

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FODE_11-15 - Tutorial FODE Ordinary First-Order...

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