國立雲林科技大學112學年度碩士班招生考試
系所: 電子系
科目: 工程數學(2)
解答: 令{P(x,y)=3x2y+6xy+y2/2Q(x,y)=3x2+y⇒{Py=3x2+6x+yQx=6x⇒Py−QxQ=1⇒取積分因子I(x)=e∫(Py−Qx)/Qdx=e∫1dx=ex⇒{I(x)P(x,y)=(3x2y+6xy+y2/2)exI(x)Q(x,y)=(3x2+y)ex⇒∂∂y(I(x)P(x,y))=∂∂y(I(x)Q(x,y))=(3x2+6x+y)ex⇒Φ(x,y)=∫I(x)P(x,y)dx+ϕ(y)=∫I(x)Q(x,y)dy+ρ(x)又{∫(3x2y+6xy+y2/2)exdx=3x2yex+12y2ex∫(3x2+y)exdy=3x2yex+12y2ex⇒ϕ(y)=ρ(x)=c⇒3x2yex+12y2ex+C=0
解答: 先求齊次解:y″+2y′+y=0⇒λ2+2λ+1=0⇒(λ+1)2=0⇒yh=c1e−x+c2xe−x,再利用變數變換法求yp{y1=e−xy2=xe−xr(x)=xe−x⇒{y′1=−e−xy′2=e−x−xe−x⇒W(y1,y2)=y1y′2−y′1y2=e−2x⇒yp=−y1∫y2rW+y2dx∫y1rWdx=−e−x∫x2dx+xe−x∫xdx=16x3e−x⇒y=yh+yp⇒y=c1e−x+c2xe−x+16x3e−x,c1及c2皆為常數
解答: (a)f(t)=(t+2)2=t2+4t+4⇒L{f(t)}=L{t2}+4L{t}+4L{1}=2s3+4s2+4s(b)L−1{F(S)}=3L−1{1S+3}+3L−1{SS2+(√5)2}=3e−3t+3cos(√5t)
解答: (a)T([x1x2x3x4])=[x1x2x3x4]⋅[2021]=2x1+2x3+x4假設u=[u1u2u3u4],v=[v1v2v3v4],則au+bv=[au1+bv1au2+bv2au3+bv3au4+bv4]⇒T(au+bv)=2(au1+bv1)+2(au3+bv3)+au4+bv4又aT(u)+bT(v)=a(2u1+2u3+u4)+b(2v1+2v3+v4)=2(au1+bv1)+2(au3+bv3)+au4+bv4=T(au+bv)⇒T(au+bv)=aT(u)+bT(v)⇒Tis linear transformation(b)[2021][x1x2x3x4]=2x1+2x3+x4=T(x)⇒A=[2021](c)kernel of A={(x1,x2,x3,x4)∣2x1+2x3+x4=0,x1,x2,x3,x4∈R}
解答: det
解答: \mathbf{(a)}\; \det(A-\lambda I)=(\lambda-0.6)(\lambda-1)=0 \Rightarrow \lambda_1=0.6,\lambda_2=1\\ \lambda_1=0.6 \Rightarrow (A-\lambda_1 I)\mathbf x=0 \Rightarrow x_1+{3\over 2}x_2=0, 取 v_1 =\begin{pmatrix}-3/2 \\1\end{pmatrix} \\ \lambda_2=1 \Rightarrow (A-\lambda_1 I)\mathbf x=0 \Rightarrow x_1+{1\over 2}x_2=0,取v_2=\begin{pmatrix}-1/2 \\1\end{pmatrix}\\ 因此D=\begin{bmatrix} \lambda_1 & 0 \\0 &\lambda_2 \end{bmatrix}\Rightarrow \bbox[red, 2pt]{D=\begin{bmatrix} 0.6 & 0 \\0 &1 \end{bmatrix}}, P=[v_1|v_2] \Rightarrow \bbox[red, 2pt]{P=\begin{bmatrix} -3/2 & -1/2 \\1 &1 \end{bmatrix}} \\\mathbf{(b)}\; \lim_{n\to \infty}A^n =\lim_{n\to \infty}(PDP^{-1})^n =\lim_{n\to \infty}PD^nP^{-1} =P\begin{bmatrix} 0 & 0 \\0 &1 \end{bmatrix}P^{-1} \\=\begin{bmatrix} -3/2 & -1/2 \\1 &1 \end{bmatrix}\begin{bmatrix} 0 & 0 \\0 &1 \end{bmatrix}\begin{bmatrix} -1 & -1/2 \\1& 3/2 \end{bmatrix}= \bbox[red,2pt]{\begin{bmatrix} -1/2 & -3/4 \\1& 3/2 \end{bmatrix}} \\ \mathbf{(c)}\;A^{-1}的特徵值={1\over \lambda_1},{1\over \lambda_2} =\bbox[red, 2pt]{{5\over 3},1}
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