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In [1]:

    
from sympy import *
init_printing()



In [2]:

    
xn=symbols('\hat{x}',real=True)
yn=Function('\hat{y}',real=True)(xn)
dsolve(yn.diff()-(yn**2-xn**2)/2/xn/yn,yn)









    Out[2]:





$$\hat{y}{\left (\hat{x} \right )} = \sqrt{\hat{x} \left(C_{1} - \hat{x}\right)}$$



In [3]:

    
c1=symbols('c1',real=True)
eps=symbols('epsilon',real=True)
xn=Function('\hat{x}',real=True)(eps)
yn=sqrt(xn*(c1-xn))
yn









    Out[3]:





$$\sqrt{\left(c_{1} - \hat{x}{\left (\epsilon \right )}\right) \hat{x}{\left (\epsilon \right )}}$$



In [4]:

    
dsolve(xn.diff()-2*xn*yn,xn)









    Out[4]:





$$\int^{\hat{x}{\left (\epsilon \right )}} \frac{1}{y \sqrt{y \left(- y + c_{1}\right)}}\, dy = C_{1} + 2 \epsilon$$

Calculamos a mano la primitiva y encontramos que es

$$\int\frac{1}{y\sqrt{y(c_1-y)}}dy=-\frac{2}{c_1}\sqrt{\frac{c_1-y}{y}}$$



In [5]:

    
c2=symbols('c2',real=True)
exp2=-2/c1*sqrt((c1-xn)/xn)
exp2









    Out[5]:





$$- \frac{2}{c_{1}} \sqrt{\frac{c_{1} - \hat{x}{\left (\epsilon \right )}}{\hat{x}{\left (\epsilon \right )}}}$$

Hay que despejar $$- \frac{2}{c_{1}} \sqrt{\frac{c_{1} - \hat{x}{\left (\epsilon \right )}}{\hat{x}{\left (\epsilon \right )}}}=c_2+2\epsilon$$

Notar que si asumimos $x>0$ entonces $c_1>0$ y $c_2+2\epsilon<0$. Pasa algo raro, sympy no sabe desdejar!!!!!! Lo resuelvo de manera media rara



In [6]:

    
(solve(exp2+2*eps+c2,xn)[0]).subs({eps:-eps,c2:-c2})









    Out[6]:





$$\frac{4 c_{1}}{c_{1}^{2} \left(- c_{2} - 2 \epsilon\right)^{2} + 4}$$



In [7]:

    
xn=4*c1/(c1**2*(-c2 - 2*eps)**2 + 4)
xn









    Out[7]:





$$\frac{4 c_{1}}{c_{1}^{2} \left(- c_{2} - 2 \epsilon\right)^{2} + 4}$$

Al $\hat{y}$ lo despejé yo, sympy no toma en consideración el problema del signo



In [8]:

    
yn=-2*c1**2*(c2 + 2*eps)/(c1**2*(c2 + 2*eps)**2 + 4)
yn









    Out[8]:





$$- \frac{2 c_{1}^{2} \left(c_{2} + 2 \epsilon\right)}{c_{1}^{2} \left(c_{2} + 2 \epsilon\right)^{2} + 4}$$



In [9]:

    
(xn.diff(eps)-2*xn*yn).simplify()









    Out[9]:





$$0$$



In [10]:

    
(yn.diff(eps)-(yn**2-xn**2)).simplify()









    Out[10]:





$$0$$



In [11]:

    
x,y=symbols('x,y',positive=True)
C1,C2=solve([xn.subs(eps,0)-x,yn.subs(eps,0)-y],[c1,c2])[0]
C1,C2









    Out[11]:





$$\left ( x + \frac{y^{2}}{x}, \quad - \frac{2 y}{x^{2} + y^{2}}\right )$$



In [12]:

    
xn=xn.subs({c1:C1,c2:C2}).simplify()
yn=yn.subs({c1:C1,c2:C2}).simplify()
xn,yn









    Out[12]:





$$\left ( \frac{x}{\epsilon^{2} x^{2} + \epsilon^{2} y^{2} - 2 \epsilon y + 1}, \quad - \frac{\left(x^{2} + y^{2}\right) \left(\epsilon \left(x^{2} + y^{2}\right) - y\right)}{x^{2} + \left(\epsilon \left(x^{2} + y^{2}\right) - y\right)^{2}}\right )$$



In [14]:

    
eps1,eps2=symbols('epsilon_1,epsilon_2',real=True)
Sus={x:xn.subs(eps,eps1),y:yn.subs(eps,eps1)}
(xn.subs(eps,eps2).subs(Sus)-xn.subs(eps,eps1+eps2)).simplify()









    Out[14]:





$$- \frac{x}{x^{2} \left(\epsilon_{1} + \epsilon_{2}\right)^{2} + y^{2} \left(\epsilon_{1} + \epsilon_{2}\right)^{2} - 2 y \left(\epsilon_{1} + \epsilon_{2}\right) + 1} + \frac{x}{\left(\epsilon_{1}^{2} x^{2} + \frac{\epsilon_{1}^{2} \left(x^{2} + y^{2}\right)^{2} \left(\epsilon_{1} \left(x^{2} + y^{2}\right) - y\right)^{2}}{\left(x^{2} + \left(\epsilon_{1} \left(x^{2} + y^{2}\right) - y\right)^{2}\right)^{2}} + \frac{2 \epsilon_{1} \left(x^{2} + y^{2}\right) \left(\epsilon_{1} \left(x^{2} + y^{2}\right) - y\right)}{x^{2} + \left(\epsilon_{1} \left(x^{2} + y^{2}\right) - y\right)^{2}} + 1\right) \left(\frac{\epsilon_{2}^{2} x^{2}}{\left(\epsilon_{1}^{2} x^{2} + \frac{\epsilon_{1}^{2} \left(x^{2} + y^{2}\right)^{2} \left(\epsilon_{1} \left(x^{2} + y^{2}\right) - y\right)^{2}}{\left(x^{2} + \left(\epsilon_{1} \left(x^{2} + y^{2}\right) - y\right)^{2}\right)^{2}} + \frac{2 \epsilon_{1} \left(x^{2} + y^{2}\right) \left(\epsilon_{1} \left(x^{2} + y^{2}\right) - y\right)}{x^{2} + \left(\epsilon_{1} \left(x^{2} + y^{2}\right) - y\right)^{2}} + 1\right)^{2}} + \frac{\epsilon_{2}^{2} \left(x^{2} + y^{2}\right)^{2} \left(\epsilon_{1} \left(x^{2} + y^{2}\right) - y\right)^{2}}{\left(x^{2} + \left(\epsilon_{1} \left(x^{2} + y^{2}\right) - y\right)^{2}\right)^{2}} + \frac{2 \epsilon_{2} \left(x^{2} + y^{2}\right) \left(\epsilon_{1} \left(x^{2} + y^{2}\right) - y\right)}{x^{2} + \left(\epsilon_{1} \left(x^{2} + y^{2}\right) - y\right)^{2}} + 1\right)}$$



In [ ]: