Question

X²y的一阶导数+xy=y²求通解

X²y的一阶导数+xy=y²求通解高数

Answer 1

求微分方程 x²y'+xy=y²的通解
解：两边同除以 x²，得：y'+(y/x)=(y/x)²........①；令y/x=u，则y=ux.......②，y'=u'x+u；
代入①式得：u'x+2u=u²，即有(du/dx)x=u(u-2)；分离变量得：du/[u(u-2)]=dx/x;
取积分得：∫du/[u(u-2)]=∫dx/x；积分之得：(1/2)∫[1/(u-2)-1/u]du=lnx+lnc₁=lnc₁x;
(1/2)[ln(u-2)-lnu]=ln√[(u-2)/u]=lnc₁x；故得√[(u-2)/u]=c₁x；(u-2)/u=cx²；(c=c₁²);
1-(2/u)=cx²; 2/u=1-cx²；u/2=1/(1-cx²)；∴u=2/(1-cx²)；
代入②式即得通解：y=2x/(1-cx²);

Answer 2

let

u = y/x
du/dx = (1/x) dy/dx - y/x^2
dy/dx = x[du/dx + (1/x)u]
/
x^2.y'+xy=y^2
y' + y/x = (y/x)^2
x[du/dx + (1/x)u] + u = u^2
xdu/dx = u^2-2u
∫du/(u^2-2u) = ∫dx/x
(1/2)∫ [1/(u-2) - 1/u ] du = ∫dx/x
(1/2)ln| (u-2)/u| = ln|x| +C'
(u-2)/u = √(Cx)
(y/x -2)/(y/x) = √(Cx)
y/x -2 = y√(C/x)
y -2x =y√(Cx)
y( 1-√(Cx) ) = 2x
y =2x/( 1-√(Cx) )