SOLDE_12-22

SOLDE_12-22 - Tutorial SOLDE: Second-Order Linear...

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Tutorial SOLDE: Second-Order Linear Differential Equations The general solution is thus y ( x ) = ® cos x + ¯ sin x + cos x 25 [(10 x +3)sin x +(5 x +4)cos x ] e ¡ 2 x + sin x 25 [(5 x +4)sin x ¡ (10 x +3)cos x ] e ¡ 2 x = ® cos x + ¯ sin x + 5 x +4 25 e ¡ 2 x : c. u 0 = ¡ sin 2 xe ¡ x = ) u = ¡ Z x sin 2 x 0 e ¡ x 0 dx 0 = 1 10 (5 +2sin2 x ¡ cos2 x ) e ¡ x : v 0 = sin x cos xe ¡ x = ) v = Z x sin x 0 cos x 0 e ¡ x 0 dx 0 = ¡ 1 10 (sin2 x +2cos2 x ) e ¡ x : The general solution is thus y ( x ) = ® cos x + ¯ sin x + cos x 10 (5+2sin2 x ¡ cos2 x ) e ¡ x ¡ sin x 10 (sin2 x +2cos2 x ) e ¡ x = ® cos x + ¯ sin x + 1 5 [2cos x + sin x ] e ¡ x : Exercise 12 y ( t ) = ® exp( m 1 t )+ ¯ exp( m 2 t ) ; _ y ( t ) = m 1 ® exp( m 1 t )+ m 2 ¯ exp( m 2 t ) ; Ä y ( t ) = m 2 1 ® exp( m 1 t )+ m 2 2 ¯ exp( m 2 t )= ) a £ m 2 1 ® exp( m 1 t )+ m 2 2 ¯ exp( m 2 t ) ¤ + b [ m 1 ® exp( m 1 t )+ m 2 ¯ exp( m 2 t )]+ c [ ® exp( m 1 t )+ ¯ exp( m 2 t )] = ® exp( m 1 t ) ¡ am 2 1 + bm 1 + c ¢ + ¯ exp( m 2 t ) ¡ am 2 2 + bm 2 + c ¢ = 0 since m 1 and m 2 are solutions to the quadratic equation, (20). 21
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Tutorial SOLDE: Second-Order Linear Differential Equations Exercise 13 Taking y 1 = e mt and ©( P )= exp μ ¡ Z t b a dt 0 = exp μ ¡ b a t ; we have from Eqn. 9, y 2 = e mt Z t e ¡ b a t 0 e 2 mt 0 dt 0 = e mt Z t exp( ¡ 2 m ¡ b=a ) t 0 dt 0 = e mt Z t dt 0 = te mt : Exercise 14 We can write m 1 = m r + m s ; m 2 = m r ¡ m s ; where m r = ¡ b= 2 a and m s = p b 2 ¡ 4 ac= 2 a: Then, Eqn. 22 becomes y ( t ) = e m r t ¡ ®e m s t + ¯e ¡ m s t ¢ = e m r t [ ® (cosh m s t + sinh m s t )+ ¯ (cosh m s t ¡ sinh m s t )] = e m r t ¡ ® 0 cosh m s t + ¯ 0 sinh m s t ¢ where ® 0 = ® + ¯ and ¯ 0 = ® ¡ ¯: Exercise 15 a. m 2 +5 m +6 =0 = ) m 1 = ¡ 2 ; m 2 = ¡ 3 = ) y ( t ) = ®e ¡ 2 t + ¯e ¡ 3 t : The conditions give 2 = ® + ¯; 0 = ¡ 2 ® ¡ 3
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This note was uploaded on 07/13/2008 for the course PHY 201 taught by Professor Covatto during the Spring '08 term at ASU.

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SOLDE_12-22 - Tutorial SOLDE: Second-Order Linear...

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