U.(P,ePr - PeP2t)QouPiz = -α± joBP0U." (Pejot - Pe-jor)-2 joαU.-"[-α(e jor -e-jor)- jo(e jor + e-jor )]-2joαsin ot+cos o ti=U,e-αtα0=cosβ0oa= U,e-α"(=ssinot+cosot)-0=sinβ00Q0oα-atU.ecos ot)sinot +0β= arctg00a00U,e-αt,sin(ot +β)V = Ke-αt sin(ot +β)V0
( ) 2 ( ) 2 1 0 2 1 2 1 0 1 2 t j t j t p t p t C e P e P e j U P e P e P P U u − − − − = − − = P12 = − j 0 [ ( ) ( )] 2 0 t j t j t j t j t e e e j e e j U − − − − − − + − = ] 2 ( ) 2 ( ) [ 0 j t j t j t j t t e e j e e U e − − − + + − = ( sin cos ) 0 U e t t t + = − ( sin cos ) 0 0 0 0 U e t t t + = − U e t V Ke t V t t sin( ) sin( ) 0 0 + = + = − − sin t cos t = = = arctg sin cos 0 0
另解uc = A,epit + A,epzt =e-αt (Aejot + A,e- jot)P-P式中 ArU.U.A二P, -PP,-PP1，P2共轭Al，Az也共轭ejot=coswt+jsinotuc =e-αt(A,ejot + A,e-jor)e-jot=coswt-jsino t=e-αt[(A, + A,)coso t+ j(A, - A,)sint)虚数实数= e-αt[Acosot+Bsinot]A=U.PtPU.=%= U,e-"[cos ot+ ~ sinot]B=J0 Ue- sin(ot +β)V同前0= Ke-αt sin(ot+β)
实数 虚数 e t j t e t j t j t j t cos sin cos sin = − = + − [(A A )cos (A A )sin ] 1 2 1 2 e t j t t = + + − − p1 ，p2 共轭 A1 ，A2 也共轭 A A (A A ) 1 2 1 2 1 2 p t p t t j t j t C u e e e e e − − = + = + = sin( + ) − Ke t t (A A ) 1 2 t j t j t C u e e e − − = + e [Acos t Bsin t] t = + − 0 2 1 2 A1 U P P P − = 0 2 1 1 A2 U P P P − − 式中 = 另解 0 0 2 1 2 1 0 U U P P P P B j A U = − + = = [cos sin ] 0 U e t t t = + − 同前 U e t V t sin( ) 0 0 + = −
uc =e-αt[Acosot+Bsinot]也可直接求A、B由初始条件duc - -αe-a'[Acos ot+Bsinot]uc(0t)=U, →A=Udt+e-at-Asinot+oBcosot]duc(0t)=0→-αA+B=0dtα00B==→A=U.U?0αuc = e-αtU,(cosot + =sinot)0α000-αtU.(或 uc =sinot)0cosot+00o0o00U.e-αt sin(wt+β)Q
(0 ) 0 A B 0 d d = → − + = + t uC 由初始条件 0 A 0 uC (0 ) = U → = U + u e [Acos t Bsin t] t C = + − A U0 B U0 = = (cos sin ) 0 u e U t t t C = + − 或 sin( ) 0 0 + = − U e t t [ A in B os ] [Acos Bsin ] d d e s t c t e t t t u t C t + − + = − + − − ( cos sin ) 0 0 0 0 u e U t t t C + = − 0 也可直接求A、B