臺灣綜合大學系統109學年度學士班轉學生聯合招生考試
科目名稱:工程數學
類組代碼:D36
解答:(1)(mx+ny)dx+(kx+ly)dy=0⇒{M(x,y)=mx+nyN(x,y)=kx+ly⇒{My=nNx=k⇒the equation is exact if n=kif n=k⇒Ψ(x,y)=∫Mdx=∫Ndy⇒{∫mx+nydx=12mx2+nxy+f(y)∫nx+lydy=nxy+12ly2+g(x)⇒Ψ(x,y)=12mx2+nxy+12ly2=C(2)my″+ny′+ky=0⇒mλ2+nλ+k=0⇒λ=−n±√n2−4mk2m=−n2m⇒yh=C1e−nx/2m+C2xe−nx/2m令yp=Aex⇒y′p=y″p=Aex⇒mAex+nAex+kAex=lex⇒A(m+n+k)=l⇒A=l/(m+n+k)⇒yp=lm+n+kex=ex⇒y=yh+yp⇒y=C1e−nx/2m+C2xe−nx/2m+ex
解答:(1)A=[1000cos(θ)−sin(θ)0sin(θ)cos(θ)]⇒{adj(A)=[1000cos(θ)sin(θ)0−sin(θ)cos(θ)]det
解答:f(x,y)={x\over y} \Rightarrow \cases{f_x= 1/y\\ f_y= -x/y^2}\\(1) (\text{grad}f) \cdot (\text{grad}f) =(f_x,f_y)\cdot (f_x, f_y) = ({1\over y},-{x\over y^2}) \cdot ({1\over y},-{x\over y^2}) = \bbox[red, 2pt]{{1\over y^2}+{x^2\over y^4}}\\ (2)\nabla^2(f^2) =\nabla^2({x^2\over y^2}) ={\partial^2\over \partial x^2 } {x^2\over y^2}+{\partial^2\over \partial y^2 } {x^2\over y^2} ={\partial\over \partial x } {2x \over y^2}+{\partial\over \partial y } {-2x^2\over y^3} =\bbox[red, 2pt]{ {2\over y^2}+ {6x^2\over y^4}}
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