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+1xy−2x⇒xy′=y2+y−2⇒1y2+y−2dy=1xdx⇒13∫(1y−1−1y+2)dy=∫1xdx⇒13(ln(y−1)−ln(y+2))=lnx+c1⇒13lny−1y+2=ln(c2x)⇒y−1y+2=c3x3⇒1−c3x3=3y+2⇒y=31−c3x3−2
解答:y″+y′+3y=0⇒λ2+λ+3=0⇒λ=−1±√11i2⇒yh=e−x/2(c1cos(√112x)+c2sin√112x)Applying undetermined coefficients, yp=Acos(2x)+Bsin(2x)⇒y′p=−2Asin(2x)+2Bcos(2x)⇒y″p=−4Acos(2x)−4Bsin(2x)⇒y″p+y′p+3yp=(−A+2B)cos(2x)+(−2A−B)sin(2x)=5sin(2x)⇒{−A+2B=0−2A−B=5⇒{A=−2B=−1⇒yp=−2cos(2x)−sin(2x)y=yh+yp⇒y=e−x/2(c1cos(√112x)+c2sin√112x)−2cos(2x)−sin(2x)

解答:L{f(t)}=L{2t2}+L{∫t0f(t−τ)e−τdτ}⇒F(s)=4s3+L{f(t)}L{e−t}=4s3+F(s)⋅1s+1⇒ss+1F(s)=4s3⇒F(s)=4(s+1)s4=4s3+4s4⇒f(t)=L−1{4s3+4s4}⇒f(t)=2t2+23t3
解答:L{y″+2y′+2y}=L{δ(t−3)}⇒s2Y(s)+2sY(s)+2Y(s)=e−3s⇒Y(s)=1s2+2s+2e−3s⇒y(t)=L−1{Y(s)}=L−1{1s2+2s+2e−3s}=L−1{1(s+1)2+1e−3s}⇒y(t)=u(t−3)e−(t−3)sin(t−3)
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