In [4]:

    

from sympy import *
init_session()


    

    




IPython console for SymPy 0.7.4.1-git (Python 3.3.3-64-bit) (ground types: python)

These commands were executed:
>>> from __future__ import division
>>> from sympy import *
>>> x, y, z, t = symbols('x y z t')
>>> k, m, n = symbols('k m n', integer=True)
>>> f, g, h = symbols('f g h', cls=Function)

Documentation can be found at http://www.sympy.org

In [5]:

    

var('theta epsilon alpha y0 y1 q')


    

    
Out[5]:





$$\begin{pmatrix}\theta, & \epsilon, & \alpha, & y_{0}, & y_{1}, & q\end{pmatrix}$$

In [6]:

    

y = exp(theta(x)/epsilon**alpha) * (y0(x) + epsilon**alpha * y1(x))
y


    

    
Out[6]:





$$\left(\epsilon^{\alpha} \operatorname{y1}{\left (x \right )} + \operatorname{y0}{\left (x \right )}\right) e^{\epsilon^{- \alpha} \theta{\left (x \right )}}$$

In [7]:

    

yp = y.diff(x).factor()
yp = yp.args[2] * expand(yp.args[0] * yp.args[1])
yp


    

    
Out[7]:





$$\left(\epsilon^{\alpha} \frac{d}{d x} \operatorname{y1}{\left (x \right )} + \operatorname{y1}{\left (x \right )} \frac{d}{d x} \theta{\left (x \right )} + \frac{d}{d x} \operatorname{y0}{\left (x \right )} + \epsilon^{- \alpha} \operatorname{y0}{\left (x \right )} \frac{d}{d x} \theta{\left (x \right )}\right) e^{\epsilon^{- \alpha} \theta{\left (x \right )}}$$

In [8]:

    

ypp = yp.diff(x).combsimp()
ypp = ypp.args[2] * expand(ypp.args[0] * ypp.args[1])
ypp


    

    
Out[8]:





$$\left(\epsilon^{\alpha} \frac{d^{2}}{d x^{2}}  \operatorname{y1}{\left (x \right )} + \operatorname{y1}{\left (x \right )} \frac{d^{2}}{d x^{2}}  \theta{\left (x \right )} + 2 \frac{d}{d x} \theta{\left (x \right )} \frac{d}{d x} \operatorname{y1}{\left (x \right )} + \frac{d^{2}}{d x^{2}}  \operatorname{y0}{\left (x \right )} + \epsilon^{- \alpha} \operatorname{y0}{\left (x \right )} \frac{d^{2}}{d x^{2}}  \theta{\left (x \right )} + \epsilon^{- \alpha} \operatorname{y1}{\left (x \right )} \left(\frac{d}{d x} \theta{\left (x \right )}\right)^{2} + 2 \epsilon^{- \alpha} \frac{d}{d x} \theta{\left (x \right )} \frac{d}{d x} \operatorname{y0}{\left (x \right )} + \epsilon^{- 2 \alpha} \operatorname{y0}{\left (x \right )} \left(\frac{d}{d x} \theta{\left (x \right )}\right)^{2}\right) e^{\epsilon^{- \alpha} \theta{\left (x \right )}}$$

In [11]:

    

f = epsilon * ypp + 2*yp + 2*y


    

In [12]:

    

f


    

    
Out[12]:





$$\epsilon \left(\epsilon^{\alpha} \frac{d^{2}}{d x^{2}}  \operatorname{y1}{\left (x \right )} + \operatorname{y1}{\left (x \right )} \frac{d^{2}}{d x^{2}}  \theta{\left (x \right )} + 2 \frac{d}{d x} \theta{\left (x \right )} \frac{d}{d x} \operatorname{y1}{\left (x \right )} + \frac{d^{2}}{d x^{2}}  \operatorname{y0}{\left (x \right )} + \epsilon^{- \alpha} \operatorname{y0}{\left (x \right )} \frac{d^{2}}{d x^{2}}  \theta{\left (x \right )} + \epsilon^{- \alpha} \operatorname{y1}{\left (x \right )} \left(\frac{d}{d x} \theta{\left (x \right )}\right)^{2} + 2 \epsilon^{- \alpha} \frac{d}{d x} \theta{\left (x \right )} \frac{d}{d x} \operatorname{y0}{\left (x \right )} + \epsilon^{- 2 \alpha} \operatorname{y0}{\left (x \right )} \left(\frac{d}{d x} \theta{\left (x \right )}\right)^{2}\right) e^{\epsilon^{- \alpha} \theta{\left (x \right )}} + 2 \left(\epsilon^{\alpha} \operatorname{y1}{\left (x \right )} + \operatorname{y0}{\left (x \right )}\right) e^{\epsilon^{- \alpha} \theta{\left (x \right )}} + 2 \left(\epsilon^{\alpha} \frac{d}{d x} \operatorname{y1}{\left (x \right )} + \operatorname{y1}{\left (x \right )} \frac{d}{d x} \theta{\left (x \right )} + \frac{d}{d x} \operatorname{y0}{\left (x \right )} + \epsilon^{- \alpha} \operatorname{y0}{\left (x \right )} \frac{d}{d x} \theta{\left (x \right )}\right) e^{\epsilon^{- \alpha} \theta{\left (x \right )}}$$

In [ ]: