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Dy yDx

详细步骤写在纸上了

如图

解:(常数变易法) 显然,y=0是方程的解。则设y≠0 ∵(x+y³)dy=ydx ∴ydx/dy=x+y³..........(1) 先解齐次方程ydx/dy=x ∵ydx/dy=x ==>dx/x=dy/y ==>ln│x│=ln│y│+ln│C│ (C是积分常数) ==>x=Cy ∴解齐次方程ydx/dy=x的通解是x=Cy (C是积分常...

解:∵(y+x)dy-ydx=0 ==>ydy+xdy-ydx=0 ==>dy/y-(ydx-xdy)/y^2=0 (等式两端同除y^2) ==>dy/y-d(x/y)=0 ==>∫dy/y-∫d(x/y)=0 ==>ln│y│-x/y=ln│C│ (C是积分常数) ==>ye^(-x/y)=C ==>y=Ce^(x/y) ∴原方程的通解是y=Ce^(x/y)。

∫2ydx+2xdy=∫2 d(xy)=2xy+C

ydx=(4x-x^2)dy dy/y=dx/(4x-x^2) 两边积分,得ln|y|=∫dx/(4x-x^2) ln|y|=-∫1/[x(x-4)]dx ln|y|=-1/4∫[1/x-1/(x-4)]dx ln|y|=-1/4[lnx-ln(x-4)]+C y=[x/(x-4)]^(1/4)*e^C

如图所示:

解:∵(x+y)dy–ydx=0 ==>ydy-(ydx-xdy)=0 ==>dy/y-(ydx-xdy)/y^2=0 (等式两端同除y^2) ==>dy/y-d(x/y)=0 ==>ln│y│-x/y=ln│C│ (C是常数) ==>ye^(-x/y)=C ==>y=Ce^(x/y) ∴原方程的通解是y=Ce^(x/y)。

求微分方程 ydx+(x²-4x)dy=0的通解 解:ydx=(4x-x²)dy 分离变量得dy/y=dx/(4x-x²) 取积分得lny=∫dx/[x(4-x)]=(1/4)∫[(1/x)+1/(4-x)]dx=(1/4)(lnx-ln(4-x)+lnc lny=(1/4)ln[x/(4-x)]+lnc=lnc[x/(4-x)]^(1/4) 故得通解:y=c[x/(4-x...

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