解答:{A(6,10,4)B(4,6,2)C(−2,−2,8)⇒{→AB=(−2,−4,−2)→AC=(−8,−12,4)⇒→n=→AB×→AC=(−40,24,−8)⇒平面方程式:−40(x−6)+24(y−10)−8(z−4)=0⇒5x−3y+z=4⇒△ABC面積=12‖→n‖=12√402+242+82=4√35
解答:y″+y=2u(t−π)−2u(t−2π)⇒L{y″}+L{y}=2L{u(t−π)}−2L{u(t−2π)}⇒s2Y(s)−sy(0)−y′(0)+Y(s)=2s(e−πs−e−2πs)⇒(s2+1)Y(s)=2s(e−πs−e−2πs)+1⇒Y(s)=2s(s2+1)(e−πs−e−2πs)+1s2+1⇒y(t)=L−1{Y(s)}⇒y(t)=2u(t−π)(1−cos(t−π))−2u(t−2π)(1−cos(t−2π))+sin(t)
解答:y″−4y′+4y=(24x2−12x)e2x=r(x)先求齊次解,y″−4y′+4y=0⇒yh=c1e2x+c2xe2x令{y1=e2xy2=xe2x⇒W(y1,y2)=|y1y2y′1y′2|=e4x⇒yp=−y1∫y2rwdx+y2∫y1rwdx=−e2x∫24x3−12x2dx+xe2x∫24x2−12xdx=2x4e2x−2x3e2xy=yh+yp⇒y=c1e2x+c2xe2x+2x4e2x−2x3e2x
解答:{M(x,y)=(x+y)2N(x,y)=2xy+x2−1⇒∂M∂y=2x+2y=∂N∂x⇒恰當(exact)方程式Φ(x,y)=∫(x+y)2dx=∫(2xy+x2−1)dy⇒13x3+x2y+xy2+ϕ(y)=xy2+x2y−y+ρ(x)⇒{ϕ(y)=−yρ(x)=x3/3⇒Φ(x,y)=13x3+x2y+xy2−y+c1=0再將y(1)=2代入⇒13+2+4−2+c1=0⇒c1=−133⇒13x3+x2y+xy2−y=133
解答:[423100420010−1−20001]−R1+R2→R2,R1/4+R3→R3→[42310000−3−1100−3/23/41/401]R1/4,R2/(−3),−2R3/3→[11/23/41/4000011/3−1/3001−1/2−1/60−2/3]−R3/2+R1→R1,−R2/2+R3→R3→[1011/301/30011/3−1/300100−1/6−2/3]−R2+R1→R1→[10001/31/30011/3−1/300100−1/6−2/3]R2↔R3→[10001/31/30100−1/6−2/30011/3−1/30]⇒A−1=[01/31/30−1/6−2/31/3−1/30]
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