匿名使用者
匿名使用者 發問時間: 教育與參考考試 · 1 0 年前

請幫我解這一份工數考題

http://www.stut.edu.tw/graduate/考古題/ML光電工程系碩士班/L光電...

只要解工數部分就好了 謝謝

至少要解一半喔 送20點

1 個解答

評分
  • 龍昊
    Lv 7
    1 0 年前
    最佳解答

    1. Find the general solution of x2y'' + 3xy' + y = 4/xsol:  Let x = et → t = ln│x│  y' = dy/dx = ( dy/dt )( dt/dx ) = ( 1/x )( dy/dt )  y'' = d2y/dx2 = ( d/dx )[ ( 1/x )( dy/dt ) ]   = ( 1/x2 )( d2y/dt2 ) - ( 1/x2 )( dy/dt )  → ( d2y/dt2 ) + 2( dy/dt ) + y = 4e-t  characteristic equation:λ2 + 2λ + 1 = 0  → ( λ + 1 )( λ + 1 ) = 0 → λ = - 1 , - 1  yh = ( c1 + c2t )e-t = ( c1 + c2ln│x│)/x  Let yp = At2e-t  → yp' = 2Ate-t - At2e-t    yp'' = At2e-t - 4Ate-t + 2Ae-t  yp'' + 2yp' + yp = 4e-t  → 2Ae-t = 4e-t → A = 2  → yp = 2t2e-t = 2‧ln2│x│/x  general solution:y = yh + yp  → y = ( c1 + c2ln│x│)/x + 2‧ln2│x│/x #3. Solve the given initial value problem by means of Laplace transforms  x' = 6x + 9y , y' = x + 6y  x( 0 ) = - 3  , y( 0 ) = - 3sol:  £{ x' }= £{ 6x + 9y }→ sX(s) + 3 = 6X(s) + 9Y(s)  → ( s - 6 )X(s) - 9Y(s) = - 3..........(1)  £{ y' }= £{ x + 6y }→ sY(s) + 3 = X(s) + 6Y(s)  → - X(s) + ( s - 6 )Y(s) = - 3..........(2)  聯立(1)、(2)兩式解 X(s)、Y(s) 得:  X(s) = ( - 3s - 9 )/[ (s - 6 )2 - 9 ]    = - 3( s - 6 )/[ (s - 6 )2 - 9 ] - 27/[ (s - 6 )2 - 9 ]  Y(s) = ( - 3s - 15 )/[ (s - 6 )2 - 9 ]  y(t) = £-1{ Y(s) }    = - 3( s - 6 )/[ (s - 6 )2 - 9 ] - 33/[ (s - 6 )2 - 9 ]  x(t) = £-1{ X(s) }  → x(t) = - e6t[3cosh(3t) + 9sinh(3t) ] #    y(t) = - e6t[3cosh(3t) + 11sinh(3t) ] #

    2005-12-31 21:33:48 補充:

    把E-mail給我吧,我把守寫稿寄給你,打字實在太辛苦了

    參考資料: 自己
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