國立 臺北科技 大學 112學 年度碩 士班招 生考試
系所組別 :1120機械工程系機電整合碩士班乙組
第一節 工程數學 試題
解答:1.令y=xm⇒y″=m(m−1)xm−2⇒4x2y″+17y=4m(m−1)xm+17xm=(4m2−4m+17)xm=0⇒4m2−4m+17=0⇒m=12±2i⇒y=c1x1/2+2i+c2x1/2−2i=c1√x⋅x2i+c2√x⋅x−2i=c1√xe2ilnx+c2√xe−2ilnx⇒y=√x(Acos(2lnx)+Bsin(2lnx))2.先求齊次解,y″+3y′+2y=0⇒λ2+3λ+2=0⇒λ=−2,−1⇒yh=c1e−2x+c2e−x令yp=ax2+bx+c⇒y′p=2ax+b⇒y″p=2a⇒y″+3y′+2y=2ax2+(6a+2b)x+2a+3b+2c=2x2⇒{2a=26a+2b=02a+3b+2c=0⇒{a=1b=−3c=7/2⇒yp=x2−3x+72⇒y=yh+yp⇒y=c1e−2x+c2e−x+x2−3x+72
解答:1.L{tcos(3t)}=−ddsL{cos(3t)}=−dds(ss2+32)=−1s2+32+2s2(s2+32)2=s2−32(s2+32)22.L{y″}+3L{y′}+2L{y}=2L{u(t−1)}⇒s2Y(s)−sy(0)−y′(0)+3(sY(s)−y(0))+2Y(s)=2⋅e−ss⇒Y(s)=2e−s(s2+3s+2)s⇒y(t)=L−1(2e−s(s2+3s+2)s)=L−1(e−s(−2s+1+1s+2+1s))=u(t−1)(−2e−(t−1)+e−2(t−1)+1)⇒y=u(t−1)(−2e−(t−1)+e−2(t−1)+1)
解答:1.A=[−4−635]⇒det
解答:\mathbf{1.}\;f(t)=\begin{cases}-1,& -\pi\lt t\lt 0\\ 1,& 0\lt t\lt \pi \end{cases} \Rightarrow f(t)為奇函數\Rightarrow a_n=0,n=0,1,2,...\\ b_n={1\over \pi}\int_{-\pi}^\pi f(x)\sin (nx)\,dx ={1\over \pi}\int_{-\pi}^0 -\sin (nx)\,dx +{1\over \pi}\int_{0}^\pi \sin (nx)\,dx \\= {1\over \pi} \left(\left.\left[ {1\over n}\cos(nx) \right] \right|_{-\pi}^0+\left.\left[ -{1\over n}\cos(nx)\right] \right|_{0}^\pi\right) ={2\over n\pi}(1-(-1)^n)=\begin{cases} 4/n\pi,& n是奇數\\ 0,& n是偶數\end{cases}\\ \Rightarrow \bbox[red,2pt]{\cases{a_n=0,n=0,1,2,\dots\\ b_n=\cases{4/n\pi,n是偶數\\ 0,n是奇數},n\in\mathbb N}} \\\mathbf{2.}\;F(\omega)=\int_{-\infty}^\infty f(t)e^{-j\omega t}\,dt =\int_{-\infty}^1 e^{2(t-1)}\;e^{-j\omega t}\,dt+\int_{1}^\infty e^{-2(t-1)}\;e^{-j\omega t}\,dt \\= e^{-2}\int_{-\infty}^1 e^{(2-j\omega)t} \,dt+e^2\int_{1}^\infty e^{-(2+j\omega)t} \,dt =e^{-2}\left. \left[ {1\over 2-j\omega}e^{(2-j\omega)t}\right] \right|_{-\infty}^1+ e^{2}\left. \left[ {1\over -2-j\omega}e^{-(2+j\omega)t} \right] \right|_{1}^\infty \\={1\over 2-j\omega}e^{-j\omega}-{1\over -2-j\omega}e^{-j\omega} =e^{-j\omega}\left({1\over 2-j\omega}+{1\over 2+j\omega}\right) =\bbox[red,2pt]{e^{-j\omega}\cdot {4\over 4+\omega^2}}
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