(dy/dx)^2-(4x)(dy/dx)+6y=0,

Question

(dy/dx)^2-(4x)(dy/dx)+6y=0,

Answer 1

令dy/dx=p(x)=p,则p^2-4xp+6y=0,
两边对x求导,得2p(dp/dx)-4p-4x(dp/dx)+6p=0,
即(p-2x) (dp/dx)+p=0,
dx/dp=2x/p-1,
解之得x=p+Cp^2,
p=[-1-(1+4Cx)^(1/2)]/(2C),或p=[-1-(1+4Cx)^(1/2)]/(2C),
即 dy/dx=[-1+(1+4Cx)^(1/2)]/(2C),
或dy/dx=[-1-(1+4Cx)^(1/2)]/(2C),
对x积分,y=-x/(2C)+(1+4Cx)^(3/2)/(12C^2)+c,
或y=-x/(2C)-(1+4Cx)^(3/2)/(12C^2)+c,
将解y的值代入原方程,求出c=-1/(12C^2),
故通解为y=-x/(2C)+[-1+(1+4Cx)^(3/2)]/(12C^2),
或y=-x/(2C)+[-1-(1+4Cx)^(3/2)]/(12C^2).