113年專門職業及技術人員高等考試
等 別:高等考試
類 科:電機工程技師
科 目:工程數學(包括線性代數、微分方程、複變函數與機率)
解答:y″−4y′+5y=0⇒λ2−4λ+5=0⇒λ=2±i⇒yh=e2x(c1cos(x)+c2sin(x))Using variations of parameters, let {y1=cos(x)e2xy2=sin(x)e2x, then W=|y1y2y′1y′2|=e4x⇒yp=−cos(x)e2x∫sin(x)e2x⋅e2xcscxe4xdx+sin(x)e2x∫cos(x)e2x⋅e2xcscxe4xdx=−cos(x)e2x∫1dx+sin(x)e2x∫cot(x)dx=−xcos(x)e2x+sin(x)e2xln(sinx)⇒y=yh+yp⇒y=e2x(c1cosx+c2sinx−xcosx+sinxln(sinx))

解答:L{f(t)}=12β3L{sin(βt)}−12β2L{tcos(βt)}⇒F(s)=12β3⋅βs2+β2−12β2⋅s2−β2(s2+β2)2⇒F(s)=1(s2+β2)2
解答:
Let f(z)=√21+16z4=√2/16z4−116=√216(z−12eπi/4)(z−12e3πi/4)(z−12e5πi/4)(z−12e7πi/4),and let C be the closed contour consisting of the interval [−R,R] on x-axis and the semicircle CR of radius R>1.∮Cf(z)dz=∫R−R√21+16x4dx+∫CRf(z)dz=I1+I2=2πi(Res(f,12eπi/4)+Res(f,12e2πi/4)),=2πi(1−4+4i+14+4i)=2πi(−14i)=π2limR→∞∫CRf(z)dz=0⇒limR→∞∫R−R√21+16x4dx=∫∞−∞√21+16x4dx=π2

解答:(一) P為對稱矩陣⇒P=[accb]⇒PA+ATP=[accb][01−5−6]+[0−51−6][accb]=−I2⇒[−10ca−5b−6ca−5b−6c2c−12b]=[−100−1]⇒{c=1/10a−5b=6c2c+1=12b⇒{a=11/10b=c=1/10⇒P=[11/101/101/101/10](二) P=[11/101/101/101/10]⇒det(P−λI)=λ2−65λ+110=0⇒特徵值λ=35±√2610
解答:(一) r(t)=[3t,4t2,8t4]⇒切線向量r′(t)=[3,8t,32t3]⇒單位切線向量r′(t)||r′(t)||=1√32+(8t)2+(32t3)2[3,8t,32t3]=1√1024t6+64t2+9[3,8t,32t3](二) By Gauss Divergence Theorem, ∬SF⋅ndA=∭div FdV=∭(7+3−1)dV=9×43π⋅33=324π
解答:(一) ∬Pdxdy=1⇒∫∞0∫∞0ke−x−y/2dxdy=∫∞0ke−y/2dy=[−2ke−y/2]|∞0=2k=1⇒k=12(二) p(y)=∫∞012e−x−y/2dx=12e−y/2⇒E{Y}=∫∞0y⋅12e−y/2dy=[−(y+2)e−y/2]|∞0=2(三) E{X2Y2}=∫∞0∫∞012x3y2e−x−y/2dxdy=∫∞012y2[−(x3+3x2+6x+6)e−x−y/2]|∞0dy=∫∞03y2e−y/2dy=3[−2(y2+4y+8)e−y/2]|∞0=3×16=48
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解題僅供參考,高普考歷年試題及詳解
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