國立臺北大學108學年度日間學士班轉學生招生
學制系級:電機三年級
科目:工程數學
解答:y2→nonlinear解答:(1)(1+lnx+yx)dx+(lnx−1)dy=0⇒{M(x,y)=1+lnx+yxN(x,y)=lnx−1⇒{∂M∂y=1x∂N∂x=1x⇒∂M∂y=∂N∂x⇒該微分方程為恰當,故得證(2)∫M(x,y)dx=∫N(x,y)dy⇒∫1+lnx+yxdx=∫lnx−1dy⇒xlnx+ylnx+ϕ(y)=ylnx−y+φ(x)+C⇒{ϕ(y)=−yφ(x)=xlnx⇒其解為ylnx+xlnx−y+C=0,C為常數
解答:令u(x)=y(x)−x⇒u′=y′−1,則原式dy−(1+ey−x+5)dx=0⇒y′=1+ey−x+5⇒u′+1=1+eu+5⇒u′=eu+5⇒∫1eudu=∫e5dx⇒−e−u=e5x+c⇒−u=ln(−e5x−c)⇒−y+x=ln(−e5x−c)⇒y=x−ln(−e5x−c),c為常數
解答:y″c−4y′c+4yc=0⇒λ2−4λ+4=0⇒(λ−2)2=0⇒yc=C1e2x+C2xe2xyp=a+bx+cx2+dx2e2x⇒y′p=b+2cx+2dxe2x+2dx2e2x⇒y″p=2c+2de2x+8dxe2x+4dx2e2x⇒y″p−4y′p+4yp=(2c+4a−4b)+(4b−8c)x+4cx2+2de2x=12+8x2−16e2x⇒{2c+4a−4b=124b=8c4c=82d=−16⇒{a=6b=4c=2d=−8⇒yp=6+4x+2x2−8x2e2x
解答:L{2+2t4+e2t+sin2t}=2L{1}+2L{t4}+L{e2t}+L{sin2t}=2s+2⋅4!s5+1s−2+2s2+22=2s+48s5+1s−2+2s2+4
解答:L−1{2s−4(s2+s)(s2+1)}=L−1{−4s+3s+1+s+3s2+1}=−4L−1{1s}+3L−1{1s+1}+L−1{ss2+1}+3L−1{1s2+1}=−4+3e−t+cost+3sint
解答:F=4xyi+(2x2+2yz)j+(3z2+y2)k=F1i+F2j+F3k⇒{F1=4xyF2=2x2+2yzF3=3z2+y2⇒curl F=|ijk∂∂x∂∂y∂∂zF1F2F3|=(∂∂yF3−∂∂zF2)i+(∂∂zF1−∂∂xF3)j+(∂∂xF2−∂∂yF1)k=0⇒div F=∂∂xF1+∂∂yF2+∂∂zF3=4y+2z+6z=4y+8z⇒{curl F=0i+0j+0kdiv F=4y+8z
解答:(1)∫c→F⋅d→r=∫c2xydx+x2dy⇒{∂∂y2xy=2x∂∂xx2=2x⇒兩者相等(exact),因此與積分路徑無關(2)因此取ϕ(x,y)滿足{ϕx=2xyϕy=x2⇒ϕ(x,y)=∫2xydx=∫x2dy⇒ϕ(x,y)=x2y(3)∫(2,4)(1,1)2xydx+x2dy=ϕ(2,4)−ϕ(1,1)=16−1=15
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