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2024年1月3日 星期三

111年台科大材料碩士班-工程數學詳解

國立臺灣科技大學111學年度碩士班招生試題

系所組別:材料科學與工程
科目:工程數學

解答:{P(x,y)=y2+xy3Q(x,y)=5y2xy+y3siny{Py=2y+3xy2Qx=yNot exactPyQxP=3y+3xy2y2+xy3=3yu=3yuu=1y3{uP=1y+xuQ=5yxy2+siny{(uP)y=1y2(uQ)x=1y2Φ(x,y)=uPdx=uQdyΦ(x,y)=xy+12x2+ϕ(y)=5lny+xycosy+ρ(x)Φ=xy+12x2+5lnycosy+C=0
解答:,x2y+xyy=0,y=xmy=mxm1y=m(m1)xm2x2y+xyy=m(m1)xm+mxmxm=(m21)xm=0m21=0m=±1y=c1x+c2x{y1=xy2=1/xW=|y1y2y1y2|=|x1/x11/x2|=2xyp=x(1/x)(1/x2(x+1))2/xdx+1xx/x2(x+1)2/xdx=x21x2(x+1)dx12x1x+1dx=x2(1xlnx+ln(x+1))12xln(x+1)=12x2lnx+x2ln(x+1)12xln(x+1)y=yh+ypy=c1x+c2x12x2lnx+x2ln(x+1)12xln(x+1)
解答:(1)3(s2+4)s(s2+4s+8)=321s+3s122(s2+4s+4)=321s+32s+2(s+2)2+2291(s+2)2+22L1{3(s2+4)s(s2+4s+8)}=32+32e2tcos(2t)9e2t12sin(2t)=32+32e2t(cos(2t)3sin(2t))(2)L{cos(3t)}=ss2+32L{tcos(3t)}=s232(s2+32)2,L{sin(3t)}=3s2+32L{sin(3t)3tcos(3t)}=3s2+323(s232)(s2+32)2=54(s2+32)2L{e2t(sin(3t)3tcos(3t))}=54((s+2)2+32)2=96(s2+4s+13)2L1{6(s2+4s+13)2}=19e2t(sin(3t)3tcos(3t))
解答:t=12ln(1x2)dtdx=x1x2d2tdx2=1+x2(1x2)2dydx=dydtdtdx=dydtx1x2d2ydx2=d2ydt2(dtdx)2+dydtd2tdx2=d2ydt2(x1x2)2+dydt1+x2(1x2)2x(1x2)2d2ydx2(1x2)2dydx+x3y=x(1x2)2(d2ydt2(x1x2)2+dydt1+x2(1x2)2)(1x2)2dydtx1x2+x3y=x3d2ydt2+2x3dydt+x3y=0d2ydt2+2dydt+y=0y=c1et+c2tet=c11x212c21x2ln(1x2)y=c11x2+c31x2ln(1x2)
解答:u(x,y)=X(x)Y(y)B.C.:{u(0,y)=0u(a,y)=0u(x,0)=0u(x,b)=f(x){X(0)Y(y)=0X(a)Y(y)=0X(x)Y(0)=0X(x)Y(b)=f(x){X(0)=0X(a)=0Y(0)=02ux2+2uy2=0XY+XY=0XX=YY=kXX=kCase I: k=0.X=0X=c1x+c2{X(0)=0X(a)=0{c2=0c1a+c2=0c1=c2=0X=0Case II: k>0.k=λ2(λ>0)X2λ2X=0X=c1eλx+c2eλx{X(0)=0X(a)=0{c1+c2=0c1eaλ+c2eaλ=0c1(e2aλ1)=0c1=0c2=0X=0Case III: k<0.k=λ2(λ>0)X2+λ2X=0X=c1cos(λx)+c2sin(λx){X(0)=c1=0X(a)=c2sin(λa)=0sin(λa)=0λa=nπλ=nπaX=c2sinnπxaYλ2Y=0Y=c3sinhλy+c4coshλy,Y(0)=0c4=0Y=c3sinhλyu=c2c3sinnπxasinhnπyau(x,y)=Ansinnπxasinhnπyau(x,y)=n=1Ansinnπxasinhnπyau(x,b)=n=1Ansinhnbπasinnπxa=f(x)Ansinhnbπa=2aa0f(x)sinnπxadxu(x,y)=n=1Ansinnπxasinhnπya,An=2asinh(nbπ/a)a0f(x)sinnπxadx


解答:f1f2dx=5π/4π/4exsinxdx=[12ex(sinxcosx)]|5π/4π/4=0Yes,f1 and f2are orthogonal to each other on[π/4,5π/4]
解答:f(x,y)=x24y2f=(fx,fy)=(2x,8y)f(2,4)=(4,32)
解答:A=[1111]det
 

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