1.xoriz.metavl
2.omogeneis.
 thetw y=z(x)*x,dy=zdx+xdz

 y'=f((a1x+b1y+g1)/(a2x+b2y+g2))
 lunw to susthma.
 1.lush: x=xo+u,y=yo+v,v(u)
  a1x+b1y+g1=a1u+b1v,..,dx=du,dy=dv,y'(x)=v'(u)
  v'(u)=f((a1u+b1v)/(a2u+b2v)) omogenhs,
  v/u=z(u) kai lunw.
 an orizousa=0 (oxi mia lush):
 b1<>0=>a1x+b1y=z(x) h
 b2<>0=>a2x+b2y=z(x)
 prwth periptwsh:
 a1+b1y'=z' a2x+b2y=l(a1x+b1y)
 z'-a1=b1*f((z+g1)/(lz+g2))
grammikes:
 yg(x)=e^(-Sfdx)[c+Sg*e^(Sfdx)dx]
Bernoulli: y'+fy=gy^p
 y^(1-p)=z(x)=>(1-p)y^-p*y'=z'
 diairoume arxikh me y^p
 antikath.tis prohgoumenes sthn
 teleftaia.->grammikh
Riccatti:y'=fy^2+gy+h
 y=y1+1/z(x),y'=y1'-z'/z^2
 prokuptei grammikh.
 me 2:y-y1/y-y3=ce^S(f(x)*y1-y2)dx
Plhreis:Pdx+Qdy=0,Py=Qx
        x         y
 F(x,y)=SP(t,y)dt+SQ(0,t)dt
        0         0
Anagomenes:
 Qx-Py/Q=g(x) =>m(x)=e^(-Sg(x)dx)
 Qx-Py/P=g(y) =>m(y)=e^(Sg(y)dy)
 h thetw p.x. z=xy kai vlepoume an
 to parakatw einai sun.mono tou z
 Qx-Py/PZy-QZx=g(z),m(z)=e^(Sg(z)dz)
Lagrange:y=xf(y')+g(y')
 1.y'(x)=p(x) y''=p'=dp/dx
 2.parag.arxikh:y'=f(y')+xf'(y')y''+g'(y')y''
 (p-f(p))dx/dp-xf'(p)=g'(p) kai lunw
 kai vriskw thn x(p).Vazw se arxikh y'=p->
 3.x(p)=G(c,p),y(p)=xf(p)+g(p)
 lush se par.morfh. Apaleifoume to p apo tis 2.
Clairaut:y=xy'+f(y')
 1.y'(x)=p(x),y''=p'=dp/dx
 2.Parag.arxikh prws x,antikathistw. opote
 vgazw koin.parag.to dp/dx
 exw dp/dx=0,x+f'(p)=0
 =>p=c->y'=c=>y=xc+f(c) apo arxikh.
 3.apo deft.kai arxikh pairnw:
 x=-f'(p),y=xp+f(p).END
