國立中正大學112學年度碩士班招生考試
科目:工程數學
系所組別: 機械工程學系光機電整合工程
解答: (a)y″+4y′=0⇒λ2+4λ=0⇒λ=0,−4⇒yh=c1+c2e−4xyp=Acosx+Bsinx+Cx+D⇒y′p=−Asinx+Bcosx+C⇒y″p=−Acosx−Bsinx⇒y″p+4y′p=(−A+4B)cosx+(−4A−B)sinx+4C=8+34cos(x)⇒{−A+4B=34−4A−B=04C=8⇒{A=−2B=8C=2⇒yp=−2cosx+8sinx+2x+D⇒y=yh+yp=c3+c2e−4x−2cosx+8sinx+2x⇒y′=−4c2e−4x+2sinx+8cosx+2{y(0)=c3+c2−2=3y′(0)=−4c2+8+2=2⇒{c2=2c3=3⇒y=3+2e−4x−2cosx+8sinx+2x(b)2y2+yexy+(4xy+xexy+2y)y′=0⇒(2y2+yexy)dx+(4xy+xexy+2y)dy=0{P(x,y)=2y2+yexyQ(x,y)=4xy+xexy+2y⇒{Py=4y+exy+xyexyQx=4y+exy+xyexy⇒Py=Qx⇒exactΦ(x,y)=∫(2y2+yexy)dx=∫(4xy+xexy+2y)dy⇒Φ(x,y)=2xy2+exy+ϕ(y)=2xy2+exy+y2+ρ(x)⇒2xy2+exy+y2+C=0

解答: A=[02−123−2−1−20]⇒det(A−λ)=−(λ+1)2(λ−5)=0⇒ eigenvalues: -1, 5, and corresponding eigenvectors: v1=[−210],v2=[101],v3=[−1−21]Applying Gram-Schmidt process, we have e1=[−2/√51/√50],e2=[1/√302/√305/√30],e3=[−1/√6−2/√61/√6]Let P=[e1∣e2∣e3]=[−2/√51/√30−1/√61/√52/√30−2/√605/√301/√6], then PPT=I and A=[−2/√51/√30−1/√61/√52/√30−2/√605/√301/√6][−1000−10005][−2/√51/√501/√302/√305/√30−1/√6−2/√61/√6]B=[5−4−2−45−2−2−28]⇒det(B−λI)=−λ(λ−9)20⇒ eigenvalues: 0,9,and corresponding eigenvectors: v1=[221],v2=[−110],v3=[−102].Applying Gram-Schmit process, we have e1=[2/32/31/3],e2=[−1/√21/√20],e3=[−√2/6−√2/62√2/3]⇒P=[e1∣e2∣e3]⇒B=[2/3−1/√2−√2/62/31/√2−√2/61/302√2/3][000090009][2/32/31/3−1/√21/√20−√2/6−√2/62√2/3]C=[22−22−14−24−1]⇒det(C−λI)=−(λ−3)2(λ+6)=0⇒eigenvalues: 3,−6and the corresponding eigenvectors: v1=[210],v2=[−201],v3=[1−22].Applying Gram-Schmit process, we have e1=[2/√51/√50],e2=[−2√5/154√5/15√5/3],e3=[1/3−2/32/3].Let P=[e1∣e2∣e3]=[2/√5−2√5/151/31/√54√5/15−2/30√5/32/3],then C=[2/√5−2√5/151/31/√54√5/15−2/30√5/32/3][30003000−6][2/√51/√50−2/√5/154√5/15√5/31/3−2/32/3]

解答: y″+3y′+2y=0⇒λ2+3λ+2=0⇒(λ+2)(λ+1)=0⇒λ=−1,−2⇒yh=c1e−x+c2e−2xyp=Acost+Bsint⇒y′p=−Asint+Bcost⇒y″p=−Acost−Bsint⇒y″p+3y′p+2yp=(A+3B)cost+(−3A+B)sint=40sint⇒{A+3B=0−3A+B=40⇒{A=−12B=4⇒yp=−12cost+4sint⇒y=yh+yp⇒y=c1e−x+c2e−2x−12cost+4sint
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