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617: 08/09(y)20:44 ID:ayZ85Z+w(1/3) AAS
(/x+a /y)f(x,y)=g(x,y)
f(x,y)=X(x)Y(y)
(/x+a /y)X(x)Y(y)=/x X(x)Y(y)+a /y X(x)Y(y)
/x X(x)Y(y)+a /y X(x)Y(y)=d/dx X(x)Y(y)+a d/dy X(x)Y(y)
(/x+a /y)f(x,y)=0???
(d/dx+a d/dy)XY=d/dx XY+a d/dy XY=0
d/dx XY=-a d/dy XY
( d/dx X)/X=-a ( d/dy Y)/Y
( d/dx X)/X=-a ( d/dy Y)/Y=
( d/dx X)/X= dX/dx=X ?1/X dX=? dx
log|X|=x+C X(x)=C_1 e^x
( d/dy Y)/Y=-/a dY/dy=-/a Y ?1/Y dY=-?/a dy
log|Y|=-/a y+C Y(y)=C_2 e^(-/a y)
f(x,y)=X(x)Y(y)=C_1 C_2 e^x e^(-/a y)=C_1 C_2 e^(/a (ax-y) )
618: 08/09(y)20:48 ID:ayZ85Z+w(2/3) AAS
?_Cf(x,y)dx
=[ab]f(x,_1(x))dx+[ba]f(x,_2(x))dx
=[ab]f(x,_1(x))dx-[ab]f(x,_2(x))dx
=-[ab]f(x,_2(x))-f(x,_1(x)) dx
=-[ab]_(_1(x))^(_2(x))(f(x,y))/y dy dx
=-_D^ (f(x,y))/y dxdy
_(_1(x))^(_2(x))(f(x,y))/y dy=[( @f(x,y)@ )]_(_1(x))^(_2(x))=f(x,_2(x))-f(x,_1(x))
619: 08/09(y)20:50 ID:ayZ85Z+w(3/3) AAS
=(/x ,/y), f=(f/x ,f/y)
(1)(C_1 f+C_2 g)=C_1 f+C_2 g
(C_1 f+C_2 g)=((C_1 f+C_2 g)/x ,(C_1 f+C_2 g)/y)
=(C_1 f/x+C_2 g/x ,C_1 f/y+C_2 g/y)
=C_1 (f/x ,f/y)+C_2 (g/x ,g/y)
(2)(fg)=(f)g+f(g)
(fg)=(fg/x ,fg/y)=(f/x g+f g/x, f/y g+f g/y)
=(f/x,f/y)g+f(g/x,g/y)=(f)g+f(g)
(3)(f/g)=((f)g-f(g))/g^2
(f/g)=(/x (f/g) ,/y (f/g))
=1/g^2 ((f/x g-f g/x) ,(f/y g-f g/y))
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