In [ ]:

    




    

In [120]:

    

x,y=symbols('x,y')
f=Function('f')(x,y)
eta1=Function('eta_1')(x)
eta2=Function('eta_2')(y)
xi=0
eta=eta1*eta2
Q=eta-eta*f
condSim=Q.diff(x)+f*Q.diff(y)-f.diff(y)*Q
condSim.simplify()


    

    
Out[120]:





$$- \eta_{1}{\left (x \right )} \eta_{2}{\left (y \right )} \frac{\partial}{\partial x} f{\left (x,y \right )} - \eta_{1}{\left (x \right )} \eta_{2}{\left (y \right )} \frac{\partial}{\partial y} f{\left (x,y \right )} - \eta_{1}{\left (x \right )} f^{2}{\left (x,y \right )} \frac{d}{d y} \eta_{2}{\left (y \right )} + \eta_{1}{\left (x \right )} f{\left (x,y \right )} \frac{d}{d y} \eta_{2}{\left (y \right )} - \eta_{2}{\left (y \right )} f{\left (x,y \right )} \frac{d}{d x} \eta_{1}{\left (x \right )} + \eta_{2}{\left (y \right )} \frac{d}{d x} \eta_{1}{\left (x \right )}$$

In [123]:

    

Q=(x+y)
EcuaQ=simplify(Q.diff(x)+f*Q.diff(y)-f.diff(y)*Q)


    

In [124]:

    

h=Function('h')(y)
sol=dsolve( EcuaQ.subs(f,h),h).rhs
sol


    

    
Out[124]:





$$C_{1} x + C_{1} y - 1$$

In [125]:

    

C1=symbols('C1')
f=sol.subs(C1,x)
f.simplify()


    

    
Out[125]:





$$x^{2} + x y - 1$$

In [129]:

    

xi=-1/x
eta=simplify(Q+f*xi)
xi,eta


    

    
Out[129]:





$$\left ( - \frac{1}{x}, \quad \frac{1}{x}\right )$$

In [167]:

    

f=x**2+x*y-1
a,b,c=symbols('a,b,c')
Q=(a*x+b*y+c)
condSim=Q.diff(x)+f*Q.diff(y)-f.diff(y)*Q
condSim=condSim.factor()
condSim,nada=fraction(condSim)
condSim


    

    
Out[167]:





$$- a x^{2} + a + b x^{2} - b - c x$$

In [156]:

    

L =[-a*c-a*d+b*c, -2*a*d,a*d-b*c-h*k,a*d]


    

    
Out[156]:





$$a d y + a k - b c y - b k - c h + d h + x^{3} \left(- a c - a d + b c\right) + x^{2} \left(- 2 a d y - a k + b k - c h - d h\right) + x \left(a d - b c - b d y^{2} - 2 d h y - h k\right)$$

In [166]:

    

condSim.subs(d,0)

L=[condSim.subs(d,0).coeff(x**i) for i in [1,2,3]]

L


    

    
Out[166]:





$$\left [ - b c - h k, \quad - a k + b k - c h, \quad - a c + b c\right ]$$

In [146]:

    

eq4=condSim.subs({x:-1,y:1})
eq4


    

    
Out[146]:





$$a c - a d - b c + b d$$

In [147]:

    

solve([eq1,eq2,eq3,eq4],[a,b,c,d])


    

    
Out[147]:





$$\left [ \left ( 0, \quad 0, \quad 0, \quad d\right ), \quad \left ( 0, \quad 0, \quad 0, \quad d\right ), \quad \left ( 0, \quad 0, \quad c, \quad d\right ), \quad \left ( 0, \quad b, \quad 0, \quad 0\right ), \quad \left ( 0, \quad b, \quad 0, \quad 0\right ), \quad \left ( a, \quad b, \quad 0, \quad 0\right )\right ]$$