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2024年3月9日 星期六

110年北科大冷凍空調碩士班-工程數學詳解

 國立臺北科技大學110學年度碩士班招生考試

系所組別:能源與冷凍空調工程
科目:工程數學

解答y=1xy2+1xy2xxy=y2+y21y2+y2dy=1xdx13(1y11y+2)dy=1xdx13(ln(y1)ln(y+2))=lnx+c113lny1y+2=ln(c2x)y1y+2=c3x31c3x3=3y+2y=31c3x32



解答y+y+3y=0λ2+λ+3=0λ=1±11i2yh=ex/2(c1cos(112x)+c2sin112x)Applying undetermined coefficients, yp=Acos(2x)+Bsin(2x)yp=2Asin(2x)+2Bcos(2x)yp=4Acos(2x)4Bsin(2x)yp+yp+3yp=(A+2B)cos(2x)+(2AB)sin(2x)=5sin(2x){A+2B=02AB=5{A=2B=1yp=2cos(2x)sin(2x)y=yh+ypy=ex/2(c1cos(112x)+c2sin112x)2cos(2x)sin(2x)


解答By method of Frobenius,x2y+5xy+(x+4)y=0x2y+xp(x)y+q(x)y=0,where p(x)=5,q(x)=4+x. indicial equation: r(r1)+p0r+q0=0r(r1)+5r+4=r24r+4=0r1=r2=2y1=n=1anxn2 and y2=y1ln(x)+n=0bnxn2x2y1+5xy1+(x+4)y1=0(a0+a1)x1+(a1+4a2)+(a2+9a3)x++(an1+n2an)xn2+=0an=1n2an1=1n2(n1)2an2==(1)n1n2(n1)212a0y1=a0n=0(1)n(n!)2xn2y2=a0n=0(1)n(n!)2xn2lnx+n=0bnxn2x2y2+5xy2+(x+4)y2=0y2=y1lnxb0x3(3+134x+3118x2+173288x3+)y=y1+y2


解答g(t)={0t<2t2+1x2g(t)=u(t2)(t2+1)L{g(t)}=L{u(t2)(t2+1)}=e2sL{(t+2)2+1}=e2s(2s3+4s2+5s)
解答L{f(t)}=L{2t2}+L{t0f(tτ)eτdτ}F(s)=4s3+L{f(t)}L{et}=4s3+F(s)1s+1ss+1F(s)=4s3F(s)=4(s+1)s4=4s3+4s4f(t)=L1{4s3+4s4}f(t)=2t2+23t3
解答L{y+2y+2y}=L{δ(t3)}s2Y(s)+2sY(s)+2Y(s)=e3sY(s)=1s2+2s+2e3sy(t)=L1{Y(s)}=L1{1s2+2s+2e3s}=L1{1(s+1)2+1e3s}y(t)=u(t3)e(t3)sin(t3)
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解題僅供參考,其他歷年試題及詳解



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