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2024年5月19日 星期日

113年成大土木碩士班-工程數學

 國立成功大學113學年度碩士班招生考試試題

系所:土木工程學系
考試科目:工程數學

解答(a)y+4y5y=0λ2+4λ5=0(λ+5)(λ1)=0λ=1,5yh=c1et+c2e5t(b)Applying variations of parameters, let {y1=ety2=e5tW=|y1y2y1y2|=6e4typ=ete5t10tet6e4tdt+e5tet10tet6e4tdt=53ette2tdt53e5tte4tdt=53et(14e2t(2t+1))53e5t(116e4t(4t1))yp=516(4t+1)et(c)y=yh+ypy=c1et+c2e5t516(4t+1)et
解答(a)A=[8332]det(AλI)=(8λ)(2λ)+9=λ210λ+25=0(λ5)2=0λ=5A has eigenvalue λ1=5 of multiplicity two.(b)c0+c1λc0c1λ1(c){f(A)=Anr(A)=c1A+c0I{f(λ)=λnr(λ)=c1λ+c0{f(λ)=r(λ)f(λ)=nλn1=r(λ)=c1{f(5)=r(5)f(5)=r(5){5n=5c1+c0n5n1=c1c0=(1n)5nAn=n5n1A+(1n)5nI=n5n1[8332]+(1n)5n[1001]=[8n5n1+(1n)5n3n5n13n5n12n5n1+(1n)5n]An=[(3n+5)5n13n5n13n5n1(53n)5n1](d)n=9A9=[3258275827582258]
解答(a)gv=yzex2+y2+z2i+xzex2+y2+z2j+xyex2+y2+z2kdiv(gv)=xyzex2+y2+z2+yxzex2+y2+z2+zxyex2+y2+z2=2xyzex2+y2+z2+2xyzex2+y2+z2+2xyzex2+y2+z2=6xyzex2+y2+z2(b)2g=2x2g+2y2g+2z2g=x2xex2+y2+z2+y2yex2+y2+z2+z2zex2+y2+z2=2ex2+y2+z2+4x2ex2+y2+z2+2ex2+y2+z2+4y2ex2+y2+z2+2ex2+y2+z2+4z2ex2+y2+z2=(6+4(x2+y2+z2))ex2+y2+z2(c)grad g=(xg,yg,zg)=(2xex2+y2+z2,2yex2+y2+z2,2zex2+y2+z2)curl(grad g)=|ijkxyz2xex2+y2+z22yex2+y2+z22zex2+y2+z2|=0(d)curl v=|ijkxyzyzxzxy|=0div(curl v)=0(e)vv=y2z2+x2z2+x2y2grad(vv)=2x(y2+z2)i+2y(x2+z2)j+2z(x2+y2)k
解答A(α)=f(x)cos(αx)dx=30cos(αx)dx=[1αsin(αx)]|30=1αsin(3α)B(α)=f(x)sin(αx)dx=30sin(αx)dx=[1αcos(αx)]|30=1α(1cos(3α)f(x)=1π0(A(α)cos(αx)+B(α)sin(αx))dα=1π0(1αsin(3α)cos(αx)+1α(1cos(3α)sin(αx))dα=1π01α(1+sin(3ααx))dα
解答A=[336224112]R2(2/3)R1R2[336048112]R3(1/3)R1R2[336048000]A=[1002/3101/301][336048000]L=[1002/3101/301],U=[336048000]
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解題僅供參考,其他歷年試題及詳解 

2 則留言:

  1. 小錯誤,糾正一下,
    2(d):A^9中的a_(2,2)=-22*5^8
    3(b):∇^2(g)=(6+4(x^2+y^2+z^2))*e^(x^2+y^2+z^2).
    謝謝!

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