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證明 X2dy-(x2+xy+y2)dx=0 (use the substitution y=vx, where v is a function of x)
Let y=x*v(x) satisfy x^2 dy - (x^2+xy+y^2) dx=0.
dy= (v+ xv') dx,
x^2 (v+xv')dx - (x^2+x^2 v+ x^2 v^2) dx=0
v' /(1+v^2) = 1/x
arctan(v)= lnx +c
y/x=v=tan(c+ lnx)
then y=x tan(c+ lnx)
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Verified answer
Let y=x*v(x) satisfy x^2 dy - (x^2+xy+y^2) dx=0.
dy= (v+ xv') dx,
x^2 (v+xv')dx - (x^2+x^2 v+ x^2 v^2) dx=0
v' /(1+v^2) = 1/x
arctan(v)= lnx +c
y/x=v=tan(c+ lnx)
then y=x tan(c+ lnx)