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2024年4月6日 星期六

112年中正大學光機電整合碩士班-工程數學詳解

國立中正大學112學年度碩士班招生考試

科目:工程數學
系所組別: 機械工程學系光機電整合工程 

解答: y+6y+5y=0yh=c1e5t+c2etUse Fourier transform to solve yp:F[y]+6F[y]+5F[y]=F[δ(t3)](ω2+6ωi+5)F[y]=eiω3F[y]=1ω2+6ωi+5e3ωi=1(ω5i)(ωi)e3ωi=14i(1ω5i1ωi)e3ωi=14(15+ωi11+ωi)e3ωiyp(t)=F1[14(15+ωi11+ωi)e3ωi]=14(e5(t3)e(t3))u(t3)y(t)=yh+ypy(t)=c1e5t+c2et14(e5(t3)e(t3))u(t3)


解答: (a)y+4y=0λ2+4λ=0λ=0,4yh=c1+c2e4xyp=Acosx+Bsinx+Cx+Dyp=Asinx+Bcosx+Cyp=AcosxBsinxyp+4yp=(A+4B)cosx+(4AB)sinx+4C=8+34cos(x){A+4B=344AB=04C=8{A=2B=8C=2yp=2cosx+8sinx+2x+Dy=yh+yp=c3+c2e4x2cosx+8sinx+2xy=4c2e4x+2sinx+8cosx+2{y(0)=c3+c22=3y(0)=4c2+8+2=2{c2=2c3=3y=3+2e4x2cosx+8sinx+2x(b)2y2+yexy+(4xy+xexy+2y)y=0(2y2+yexy)dx+(4xy+xexy+2y)dy=0{P(x,y)=2y2+yexyQ(x,y)=4xy+xexy+2y{Py=4y+exy+xyexyQx=4y+exy+xyexyPy=QxexactΦ(x,y)=(2y2+yexy)dx=(4xy+xexy+2y)dyΦ(x,y)=2xy2+exy+ϕ(y)=2xy2+exy+y2+ρ(x)2xy2+exy+y2+C=0

解答: A[021232120]det(Aλ)=(λ+1)2(λ5)=0 eigenvalues: -1, 5, and corresponding eigenvectors: v1=[210],v2=[101],v3=[121]Applying Gram-Schmidt process, we have e1=[2/51/50],e2=[1/302/305/30],e3=[1/62/61/6]Let P=[e1e2e3]=[2/51/301/61/52/302/605/301/6], then PPT=I and A=[2/51/301/61/52/302/605/301/6][100010005][2/51/501/302/305/301/62/61/6]B=[542452228]det(BλI)=λ(λ9)20 eigenvalues: 0,9,and corresponding eigenvectors: v1=[221],v2=[110],v3=[102].Applying Gram-Schmit process, we have e1=[2/32/31/3],e2=[1/21/20],e3=[2/62/622/3]P=[e1e2e3]B=[2/31/22/62/31/22/61/3022/3][000090009][2/32/31/31/21/202/62/622/3]C=[222214241]det(CλI)=(λ3)2(λ+6)=0eigenvalues: 3,6and the corresponding eigenvectors: v1=[210],v2=[201],v3=[122].Applying Gram-Schmit process, we have e1=[2/51/50],e2=[25/1545/155/3],e3=[1/32/32/3].Let P=[e1e2e3]=[2/525/151/31/545/152/305/32/3],then C=[2/525/151/31/545/152/305/32/3][300030006][2/51/502/5/1545/155/31/32/32/3]

解答: (a)L{y}+4L{y}=s2Y(s)sy(0)y(0)+4Y(s)=(s2+4)Y(s)+8=0Y(s)=8s2+4y(t)=L1{Y(s)}=L1{42s2+22}=4sin(2t)y(t)=4sin(2t)(b)L{y}+2L{y}L{y}=s2Y(s)6+2sY(s)Y(s)=0Y(s)=6s2+2s1y(t)=L1{Y(s)}=L1{6(s+1)22}=32etsinh(2t)y(t)=32etsinh(2t)

解答: y+3y+2y=0λ2+3λ+2=0(λ+2)(λ+1)=0λ=1,2yh=c1ex+c2e2xyp=Acost+Bsintyp=Asint+Bcostyp=AcostBsintyp+3yp+2yp=(A+3B)cost+(3A+B)sint=40sint{A+3B=03A+B=40{A=12B=4yp=12cost+4sinty=yh+ypy=c1ex+c2e2x12cost+4sint

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解題僅供參考,其他歷年試題及詳解

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