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2024年1月30日 星期二

111年成大資源工程碩士班-工程數學詳解

 國立成功大學111學年度碩士班招生考試

系所:資源工程學系
科目:工程數學

解答:(a)y+y=0yh=c1cosx+c2sinxyp=Acos(2x)+Bsin(2x)yp=2Asin(2x)+2Bcos(2x)yp=4Acos(2x)4Bsin(2x)yp+yp=3Acos(2x)3Bsin(2x)=sin(2x){A=0B=1/3yp=13sin(2x)y=yh+ypy=c1cosx+c2sinx13sin(2x)y=c1sinx+c2cosx23cos(2x){y(0)=c1=0y(0)=c22/3=0c2=23y=23sinx23sin(2x)(b)y=y+exyy=ex integration factor I(x)=e1dx=exexyexy=1(exy)=1exy=x+c1y=xex+c1exy(0)=c1=2y=xex+2ex

解答:(a)0e3t+2estdt=e20e(3s)tdt=e2[13se(3s)t]|0=e2s3(b)L1{s+6s2+4s+20}=L1{(s+2)+4(s+2)2+42}=e2t(cos4t+sin4t)


解答:(a-1)[1342211312311301]R22R1R2,R3R1R3,R4R1R4[1342077701110041]R1+3R3R1,R27R3R2[1011000001110041]R3R3,R4/4R4[1011000001110011/4]R1R4R1,R2R3[1005/4011100000011/4]R2R4R2,R3R4[1005/40103/40011/40000](a-2)rref(A)rank(A)3(a-3)rank(A)=11+3+1=4(b)B=[010100001]det(BλI)=(λ1)2(λ+1)=0eigenvalues: 1,1λ1=1(Bλ1I)v=0[110110000][x1x2x3]=0x1=x2v=x1(110)+x3(001),v1=(110),v2=(001)λ2=1(Bλ1I)v=0[110110002][x1x2x3]=0{x1+x2=0x3=0v=x2(110),v3=(110)eigenvectors: (110),(001),(110)


解答:(a)F(x,y,z)=(3xy2,2yz2,4x2z)div F=x3xy2+y(2yz2)+z4x2z=3y22z2+4x2div F(1,2,3)=1218+4=2(b){F=x+y2+z3v=(2,2,1){(Fx,Fy,Fz)=(1,2y,3z2)n=vv=(23,23,13)DnF(x,y,z)=(1,2y,3z2)(23,23,13)=23+4y3+z2DnF(3,2,1)=23+83+1=133(c)C,R,P(x,y)Q(x,y),CPdx+Qdy=
解答:\textbf{(a)} \; f(x)為奇函數 \Rightarrow a_n=0,而b_n= \int_{-1}^0 -\sin(n\pi x)\,dx + \int_0^1 \sin(n\pi x)\,dx ={2\over n\pi}(1-(-1)^n)\\\quad  \Rightarrow f(x)= \bbox[red, 2pt]{\sum_{n=1}^\infty {2\over n\pi}(1-(-1)^n) \sin(n\pi x)} \\\textbf{(a)} \; f(x)=e^{-x}, x\gt 0 \Rightarrow A(\alpha)=\int_0^\infty e^{-x} \cos(\alpha x)\,dx ={1\over 1+\alpha^2} \\ \quad \Rightarrow \bbox[red, 2pt]{ f(x)={2\over \pi}\int_0^\infty {\cos (\alpha x)\over 1+\alpha^2} \,dx} \\\textbf{(c)} \; F(\omega) =\int_{-\infty}^\infty e^{-|x|} e^{-j\omega x}\,dx =2\int_0^\infty e^{-x} e^{-j\omega x}\,dx = 2\int_0^\infty   e^{-(j\omega+1) x}\,dx =2 \left. \left[ -{1\over j\omega +1} e^{-(j\omega+1)x}\right] \right|_0^\infty\\ \qquad =\bbox[red, 2pt]{2\over j\omega +1}

解答:\text{for example, RLC circuit}

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解題僅供參考,其他歷年試題及詳解


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