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2022年12月1日 星期四

107年台綜大轉學考-工程數學D09詳解

臺灣綜合大學系統107學年度學士班轉學生聯合招生考試

科目名稱:工程數學
類組代碼:D09




解答:yh+yh=0yh=Acosx+Bsinxr(x)=4x+10sinxy=yh+yp=ax+b+Acosx+Bsinx+Cxcosx+Dxsinxy=a+(DA)sinx+(B+C)cosxCxsinx+Dxcosxy=(2DA)cosx(B+2C)sinxCxcosxDxsinxy+y=ax+b+2Dcosx2Csinx=4x+10sinx{a=4b=0C=5D=0y=4x+Acosx+Bsinx5xcosxy=4+(5A)sinx+(B5)cosx{y(π)=0y(π)=2{4πA+5π=04+5B=2{A=9πB=7y=4x+9πcosx+7sinx5xcosx
解答(a)L{cos(5t)}=ss2+52=ss2+25(b)L1{1s1}=et(c)L{y}L{y}=2L{cos(5t)}sY(s)y(0)Y(s)=2ss2+25Y(s)=2s(s2+25)(s1)=113s+2513s2+25+1/13s1=113ss2+25+25131s2+25+1131s1y(t)=113L1{ss2+25}+513L1{5s2+25}+113L1{1s1}=113cos(5t)+513sin(5t)+113ety(t)=113cos(5t)+513sin(5t)+113et

解答(a)A=[010100001]det(AλI)=0|λ101λ0001λ|=(λ1)2(λ+1)=0A1,1(b)λ1=1(Aλ1I)X=0[110110000][x1x2x3]=0x1=x2v1=[1/21/20],v2=[001]λ2=1(Aλ2I)X=0[110110002][x1x2x3]=0{x1=x2x3=0v3=[1/21/20](orthonormal eigen vector)[1/21/20],[001],[1/21/20](c)A=[010100001]=[101101010][100010001][101101010]1=P[100010001]P1sin(A)=P[sin(1)000sin(1)000sin(1)]P1sin(A3)=P[sin3(1)000sin3(1)000sin3(1)]P1sin(3A)=3sin(A)4sin3(A)=P[3sin(1)4sin3(1)0003sin(1)4sin3(1)0003sin(1)+4sin3(1)]P1=[101101010][3sin(1)4sin3(1)0003sin(1)4sin3(1)0003sin(1)+4sin3(1)][1/21/200011/21/20]=[3sin(1)4sin3(1)03sin(1)4sin3(1)3sin(1)4sin3(1)03sin(1)+4sin3(1)03sin(1)4sin3(1)0][1/21/200011/21/20]=[06sin(1)8sin3(1)06sin(1)8sin3(1)00003sin(1)4sin3(1)]


解答

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解題僅供參考,其他歷年試題及詳解


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