In [1]:

    

from sympy import *
init_printing()


    

In [74]:

    

t=symbols('t')
x=Function('x')(t)
y=Function('y')(t)
z=Function('z')(t)
ec1=Eq(x.diff(t),x+z)
ec2=Eq(y.diff(t),x*t-z)
ec3=Eq(z.diff(t),y+x*t**2)
Matrix([ec1, ec2, ec3])


    

    
Out[74]:





$$\left[\begin{matrix}\frac{d}{d t} x{\left (t \right )} = x{\left (t \right )} + z{\left (t \right )}\\\frac{d}{d t} y{\left (t \right )} = t x{\left (t \right )} - z{\left (t \right )}\\\frac{d}{d t} z{\left (t \right )} = t^{2} x{\left (t \right )} + y{\left (t \right )}\end{matrix}\right]$$

In [75]:

    

ec21=Eq(ec1.lhs.diff(t),ec1.rhs.diff(t))
ec21.rhs.subs({x.diff(t):ec1.rhs,y.diff(t):ec2.rhs,z.diff(t):ec3.rhs})
ec21=Eq(ec1.lhs.diff(t),ec21.rhs.subs({x.diff(t):ec1.rhs,y.diff(t):ec2.rhs,z.diff(t):ec3.rhs}))
ec21


    

    
Out[75]:





$$\frac{d^{2}}{d t^{2}}  x{\left (t \right )} = t^{2} x{\left (t \right )} + x{\left (t \right )} + y{\left (t \right )} + z{\left (t \right )}$$

In [76]:

    

ec31=Eq(ec21.lhs.diff(t),ec21.rhs.diff(t))
ec31
ec31.rhs.subs({x.diff(t):ec1.rhs,y.diff(t):ec2.rhs,z.diff(t):ec3.rhs})
ec31=Eq(ec21.lhs.diff(t),ec31.rhs.subs({x.diff(t):ec1.rhs,y.diff(t):ec2.rhs,z.diff(t):ec3.rhs}))
ec31.simplify()


    

    
Out[76]:





$$\frac{d^{3}}{d t^{3}}  x{\left (t \right )} = 2 t^{2} x{\left (t \right )} + t^{2} z{\left (t \right )} + 3 t x{\left (t \right )} + x{\left (t \right )} + y{\left (t \right )}$$

In [77]:

    

solve([ec1,ec21,ec31],[x,y,z])


    

    
Out[77]:





$$\left \{ x{\left (t \right )} : \frac{1}{3 t + 1} \left(- \left(t^{2} - 1\right) \frac{d}{d t} x{\left (t \right )} - \frac{d^{2}}{d t^{2}}  x{\left (t \right )} + \frac{d^{3}}{d t^{3}}  x{\left (t \right )}\right), \quad y{\left (t \right )} : \frac{1}{3 t + 1} \left(t^{4} \frac{d}{d t} x{\left (t \right )} - t^{2} \frac{d}{d t} x{\left (t \right )} + t^{2} \frac{d^{2}}{d t^{2}}  x{\left (t \right )} - t^{2} \frac{d^{3}}{d t^{3}}  x{\left (t \right )} - 3 t \frac{d}{d t} x{\left (t \right )} + 3 t \frac{d^{2}}{d t^{2}}  x{\left (t \right )} - \frac{d}{d t} x{\left (t \right )} + \frac{d^{2}}{d t^{2}}  x{\left (t \right )}\right), \quad z{\left (t \right )} : \frac{1}{3 t + 1} \left(t \left(t + 3\right) \frac{d}{d t} x{\left (t \right )} + \frac{d^{2}}{d t^{2}}  x{\left (t \right )} - \frac{d^{3}}{d t^{3}}  x{\left (t \right )}\right)\right \}$$

In [ ]:

    




    

In [88]:

    

t=symbols('t')
xx=Function('xx')(t)
yy=Function('yy')(t)

ec1=Eq(xx.diff(t),xx+yy)
ec2=Eq(yy.diff(t),xx*yy)

Matrix([ec1, ec2])


    

    
Out[88]:





$$\left[\begin{matrix}\frac{d}{d t} \operatorname{xx}{\left (t \right )} = \operatorname{xx}{\left (t \right )} + \operatorname{yy}{\left (t \right )}\\\frac{d}{d t} \operatorname{yy}{\left (t \right )} = \operatorname{xx}{\left (t \right )} \operatorname{yy}{\left (t \right )}\end{matrix}\right]$$

In [89]:

    

ec21=Eq(ec1.lhs.diff(t),ec1.rhs.diff(t))
ec21.rhs.subs({xx.diff(t):ec1.rhs,yy.diff(t):ec2.rhs})
ec21=Eq(ec1.lhs.diff(t),ec21.rhs.subs({xx.diff(t):ec1.rhs,yy.diff(t):ec2.rhs}))
ec21


    

    
Out[89]:





$$\frac{d^{2}}{d t^{2}}  \operatorname{xx}{\left (t \right )} = \operatorname{xx}{\left (t \right )} \operatorname{yy}{\left (t \right )} + \operatorname{xx}{\left (t \right )} + \operatorname{yy}{\left (t \right )}$$

In [90]:

    

solve([ec1,ec21],[xx,yy])


    

    
Out[90]:





$$\left [ \left ( - \frac{1}{2} \sqrt{\frac{d}{d t}\left(- \frac{1}{2} \sqrt{\frac{d}{d t} \operatorname{xx}{\left (t \right )}^{2} + 4 \frac{d}{d t} \operatorname{xx}{\left (t \right )} - 4 \frac{d^{2}}{d t^{2}}  \operatorname{xx}{\left (t \right )}} + \frac{1}{2} \frac{d}{d t} \operatorname{xx}{\left (t \right )}\right)^{2} + 4 \frac{d}{d t}\left(- \frac{1}{2} \sqrt{\frac{d}{d t} \operatorname{xx}{\left (t \right )}^{2} + 4 \frac{d}{d t} \operatorname{xx}{\left (t \right )} - 4 \frac{d^{2}}{d t^{2}}  \operatorname{xx}{\left (t \right )}} + \frac{1}{2} \frac{d}{d t} \operatorname{xx}{\left (t \right )}\right) - 4 \frac{d^{2}}{d t^{2}} \left(- \frac{1}{2} \sqrt{\frac{d}{d t} \operatorname{xx}{\left (t \right )}^{2} + 4 \frac{d}{d t} \operatorname{xx}{\left (t \right )} - 4 \frac{d^{2}}{d t^{2}}  \operatorname{xx}{\left (t \right )}} + \frac{1}{2} \frac{d}{d t} \operatorname{xx}{\left (t \right )}\right)} + \frac{1}{2} \frac{d}{d t}\left(- \frac{1}{2} \sqrt{\frac{d}{d t} \operatorname{xx}{\left (t \right )}^{2} + 4 \frac{d}{d t} \operatorname{xx}{\left (t \right )} - 4 \frac{d^{2}}{d t^{2}}  \operatorname{xx}{\left (t \right )}} + \frac{1}{2} \frac{d}{d t} \operatorname{xx}{\left (t \right )}\right), \quad \frac{1}{2} \sqrt{\frac{d}{d t}\left(- \frac{1}{2} \sqrt{\frac{d}{d t} \operatorname{xx}{\left (t \right )}^{2} + 4 \frac{d}{d t} \operatorname{xx}{\left (t \right )} - 4 \frac{d^{2}}{d t^{2}}  \operatorname{xx}{\left (t \right )}} + \frac{1}{2} \frac{d}{d t} \operatorname{xx}{\left (t \right )}\right)^{2} + 4 \frac{d}{d t}\left(- \frac{1}{2} \sqrt{\frac{d}{d t} \operatorname{xx}{\left (t \right )}^{2} + 4 \frac{d}{d t} \operatorname{xx}{\left (t \right )} - 4 \frac{d^{2}}{d t^{2}}  \operatorname{xx}{\left (t \right )}} + \frac{1}{2} \frac{d}{d t} \operatorname{xx}{\left (t \right )}\right) - 4 \frac{d^{2}}{d t^{2}} \left(- \frac{1}{2} \sqrt{\frac{d}{d t} \operatorname{xx}{\left (t \right )}^{2} + 4 \frac{d}{d t} \operatorname{xx}{\left (t \right )} - 4 \frac{d^{2}}{d t^{2}}  \operatorname{xx}{\left (t \right )}} + \frac{1}{2} \frac{d}{d t} \operatorname{xx}{\left (t \right )}\right)} + \frac{1}{2} \frac{d}{d t}\left(- \frac{1}{2} \sqrt{\frac{d}{d t} \operatorname{xx}{\left (t \right )}^{2} + 4 \frac{d}{d t} \operatorname{xx}{\left (t \right )} - 4 \frac{d^{2}}{d t^{2}}  \operatorname{xx}{\left (t \right )}} + \frac{1}{2} \frac{d}{d t} \operatorname{xx}{\left (t \right )}\right)\right ), \quad \left ( \frac{1}{2} \sqrt{\frac{d}{d t}\left(\frac{1}{2} \sqrt{\frac{d}{d t} \operatorname{xx}{\left (t \right )}^{2} + 4 \frac{d}{d t} \operatorname{xx}{\left (t \right )} - 4 \frac{d^{2}}{d t^{2}}  \operatorname{xx}{\left (t \right )}} + \frac{1}{2} \frac{d}{d t} \operatorname{xx}{\left (t \right )}\right)^{2} + 4 \frac{d}{d t}\left(\frac{1}{2} \sqrt{\frac{d}{d t} \operatorname{xx}{\left (t \right )}^{2} + 4 \frac{d}{d t} \operatorname{xx}{\left (t \right )} - 4 \frac{d^{2}}{d t^{2}}  \operatorname{xx}{\left (t \right )}} + \frac{1}{2} \frac{d}{d t} \operatorname{xx}{\left (t \right )}\right) - 4 \frac{d^{2}}{d t^{2}} \left(\frac{1}{2} \sqrt{\frac{d}{d t} \operatorname{xx}{\left (t \right )}^{2} + 4 \frac{d}{d t} \operatorname{xx}{\left (t \right )} - 4 \frac{d^{2}}{d t^{2}}  \operatorname{xx}{\left (t \right )}} + \frac{1}{2} \frac{d}{d t} \operatorname{xx}{\left (t \right )}\right)} + \frac{1}{2} \frac{d}{d t}\left(\frac{1}{2} \sqrt{\frac{d}{d t} \operatorname{xx}{\left (t \right )}^{2} + 4 \frac{d}{d t} \operatorname{xx}{\left (t \right )} - 4 \frac{d^{2}}{d t^{2}}  \operatorname{xx}{\left (t \right )}} + \frac{1}{2} \frac{d}{d t} \operatorname{xx}{\left (t \right )}\right), \quad - \frac{1}{2} \sqrt{\frac{d}{d t}\left(\frac{1}{2} \sqrt{\frac{d}{d t} \operatorname{xx}{\left (t \right )}^{2} + 4 \frac{d}{d t} \operatorname{xx}{\left (t \right )} - 4 \frac{d^{2}}{d t^{2}}  \operatorname{xx}{\left (t \right )}} + \frac{1}{2} \frac{d}{d t} \operatorname{xx}{\left (t \right )}\right)^{2} + 4 \frac{d}{d t}\left(\frac{1}{2} \sqrt{\frac{d}{d t} \operatorname{xx}{\left (t \right )}^{2} + 4 \frac{d}{d t} \operatorname{xx}{\left (t \right )} - 4 \frac{d^{2}}{d t^{2}}  \operatorname{xx}{\left (t \right )}} + \frac{1}{2} \frac{d}{d t} \operatorname{xx}{\left (t \right )}\right) - 4 \frac{d^{2}}{d t^{2}} \left(\frac{1}{2} \sqrt{\frac{d}{d t} \operatorname{xx}{\left (t \right )}^{2} + 4 \frac{d}{d t} \operatorname{xx}{\left (t \right )} - 4 \frac{d^{2}}{d t^{2}}  \operatorname{xx}{\left (t \right )}} + \frac{1}{2} \frac{d}{d t} \operatorname{xx}{\left (t \right )}\right)} + \frac{1}{2} \frac{d}{d t}\left(\frac{1}{2} \sqrt{\frac{d}{d t} \operatorname{xx}{\left (t \right )}^{2} + 4 \frac{d}{d t} \operatorname{xx}{\left (t \right )} - 4 \frac{d^{2}}{d t^{2}}  \operatorname{xx}{\left (t \right )}} + \frac{1}{2} \frac{d}{d t} \operatorname{xx}{\left (t \right )}\right)\right )\right ]$$