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)=5y2−xy+y3siny⇒{Py=2y+3xy2Qx=−y⇒Not exact−Py−QxP=−3y+3xy2y2+xy3=−3y⇒u′=−3yu⇒u=1y3⇒{uP=1y+xuQ=5y−xy2+siny⇒{(uP)y=−1y2(uQ)x=−1y2⇒Φ(x,y)=∫uPdx=∫uQdy⇒Φ(x,y)=xy+12x2+ϕ(y)=5lny+xy−cosy+ρ(x)⇒Φ=xy+12x2+5lny−cosy+C=0解答:先求齊次解,x2y″+xy′−y=0,令y=xm⇒y′=mxm−1⇒y″=m(m−1)xm−2⇒x2y″+xy′−y=m(m−1)xm+mxm−xm=(m2−1)xm=0⇒m2−1=0⇒m=±1⇒y=c1x+c2x令{y1=xy2=1/x⇒W=|y1y2y′1y′2|=|x1/x1−1/x2|=−2x⇒yp=−x∫(1/x)⋅(1/x2(x+1))−2/xdx+1x∫x/x2(x+1)−2/xdx=x2∫1x2(x+1)dx−12x∫1x+1dx=x2(−1x−lnx+ln(x+1))−12xln(x+1)=−12−x2lnx+x2ln(x+1)−12xln(x+1)⇒y=yh+yp⇒y=c1x+c2x−12−x2lnx+x2ln(x+1)−12xln(x+1)
解答:(1)3(s2+4)s(s2+4s+8)=32⋅1s+3s−122(s2+4s+4)=32⋅1s+32⋅s+2(s+2)2+22−9⋅1(s+2)2+22⇒L−1{3(s2+4)s(s2+4s+8)}=32+32e−2tcos(2t)−9e−2t⋅12sin(2t)=32+32e−2t(cos(2t)−3sin(2t))(2)L{cos(3t)}=ss2+32⇒L{tcos(3t)}=s2−32(s2+32)2,又L{sin(3t)}=3s2+32⇒L{sin(3t)−3tcos(3t)}=3s2+32−3(s2−32)(s2+32)2=54(s2+32)2⇒L{e−2t(sin(3t)−3tcos(3t))}=54((s+2)2+32)2=9⋅6(s2+4s+13)2⇒L−1{6(s2+4s+13)2}=19e−2t(sin(3t)−3tcos(3t))
解答:t=−12ln(1−x2)⇒dtdx=x1−x2⇒d2tdx2=1+x2(1−x2)2又dydx=dydtdtdx=dydt⋅x1−x2⇒d2ydx2=d2ydt2(dtdx)2+dydtd2tdx2=d2ydt2(x1−x2)2+dydt1+x2(1−x2)2原式x(1−x2)2d2ydx2−(1−x2)2dydx+x3y=x(1−x2)2(d2ydt2(x1−x2)2+dydt1+x2(1−x2)2)−(1−x2)2dydt⋅x1−x2+x3y=x3d2ydt2+2x3dydt+x3y=0⇒d2ydt2+2dydt+y=0⇒y=c1e−t+c2te−t=c1√1−x2−12c2√1−x2ln(1−x2)⇒y=c1√1−x2+c3√1−x2ln(1−x2)
解答:取u(x,y)=X(x)Y(y)則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)=0而原式∂2u∂x2+∂2u∂y2=0⇒X″Y+XY″=0⇒X″X=−Y″Y=k先討論X″X=kCase I: k=0.X″=0⇒X=c1x+c2⇒{X(0)=0X(a)=0⇒{c2=0c1a+c2=0⇒c1=c2=0⇒X=0Case II: k>0.假設k=λ2(λ>0)⇒X2−λ2X=0⇒X=c1eλx+c2e−λx⇒{X(0)=0X(a)=0⇒{c1+c2=0c1eaλ+c2e−aλ=0⇒c1(e2aλ−1)=0⇒c1=0⇒c2=0⇒X=0Case III: k<0.假設k=−λ2(λ>0)⇒X2+λ2X=0⇒X=c1cos(λx)+c2sin(λx){X(0)=c1=0X(a)=c2sin(λa)=0⇒sin(λa)=0⇒λa=nπ⇒λ=nπa⇒X=c2sinnπxaY″−λ2Y=0⇒Y=c3sinhλy+c4coshλy,又Y(0)=0⇒c4=0⇒Y=c3sinhλy⇒u=c2c3sinnπxasinhnπya⇒u(x,y)=Ansinnπxasinhnπya⇒u(x,y)=∞∑n=1Ansinnπxasinhnπya⇒u(x,b)=∞∑n=1Ansinhnbπasinnπxa=f(x)⇒Ansinhnbπa=2a∫a0f(x)sinnπxadx⇒u(x,y)=∞∑n=1Ansinnπxasinhnπya,其中An=2asinh(nbπ/a)∫a0f(x)sinnπxadx
解答:∫f1f2dx=∫5π/4π/4exsinxdx=[12ex(sinx−cosx)]|5π/4π/4=0⇒Yes,f1 and f2are orthogonal to each other on[π/4,5π/4]
解答:f(x,y)=x2−4y2⇒∇f=(fx,fy)=(2x,−8y)⇒∇f(2,4)=(4,−32)
解答:A=[1−111]⇒det
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