Final Example

Computer Algebra or Symbolic Computing is particularly useful when a problem involves a large amount of very tedious work where it is easy to miss or lose important terms. One such is the solution of non-linear differential equations using a series expansion. We therefore illustrate such a solution where we are trying to solve the equation

$${\d y\over \d x} = a y^2.$$ (5.1)

In[40]:=	y = a0 + a1 x + a2 x^2 + a3 x^3 + a4 x^4
Out[40]=	$${a0 + a1\ x + a2\ x^2 + a3\ x^3 + a4\ x^4}$$
In[41]:=	D[y, x] - a (y)^2
Out[41]=	$${a1 + 2\ a2\ x + 3\ a3\ x^2 + 4\ a4\ x^3}$$
	$- a (a0 + a1\ x + a2\ x^2 + a3\ x^3 + a4\ x^4)^2$
In[42]:=	equ = Expand[%]
Out[42]=	$${-(a\ a0^2) + a1 - 2\ a\ a0\ a1\ x + 2\ a2\ x - a\ a1^2\ x^2 - 2\ a\ a0\ a2\ x^2}$$
	$+ 3\ a3\ x^2 - 2\ a\ a1\ a2\ x^3 - 2\ a\ a0\ a3\ x^3 + 4\ a4\ x^3$
	$- a\ a2^2\ x^4 - 2\ a\ a1\ a3\ x^4 - 2\ a\ a0\ a4\ x^4 - 2\ a\ a2\ a3\ x^5$
	$- 2\ a\ a1\ a4\ x^5 - a\ a3^2\ x^6 - 2\ a\ a2\ a4\ x^6 - 2\ a\ a3\ a4\ x^7$
	$- a\ a4^2\ x^8$
In[43]:=	equ/.x $\mathrel{-\negthinspace\negthinspace>}$0
Out[43]=	$${-(a\ a0)^2 + a1}$$
In[44]:=	Solve[%==0, a1]
Out[44]=	$${\{\{a1\mathrel{-\negthinspace\negthinspace>}a\ a0^2\}\}}$$
In[45]:=	a1 = a1/.%[[1]]
Out[45]=	$${a\ a0^2}$$

In[46]:=	equ
Out[46]=	$${-2\ a^2\ a0^3\ x + 2\ a2\ x - a^3\ a0^4\ x^2 - 2\ a\ a0\ a2\ x^2 + 3\ a3\ x^2}$$
	$- 2\ a^2\ a0^2\ a2\ x^3 - 2\ a\ a0\ a3\ x^3 + 4\ a4\ x^3 - a\ a2^2\ x^4$
	$- 2\ a^2\ a0^2\ a3\ x^4 - 2\ a\ a0\ a4\ x^4 - 2\ a\ a2\ a3\ x^5 - 2\ a^2\ a0^2\ a4\ x^5$
	$- a\ a3^2\ x^6 - 2\ a\ a2\ a4\ x^6 - 2\ a\ a3\ a4\ x^7 - a\ a4^2\ x^8$
In[47]:=	Coefficient[equ, x]
Out[47]=	$${-2\ a^2\ a0^3 + 2\ a2}$$
In[48]:=	Solve[%==0, a2]
Out[48]=	$${\{\{a2\mathrel{-\negthinspace\negthinspace>}a^2\ a0^3\}\} }$$
In[49]:=	a2 = a2/.%[[1]]
Out[49]=	$${a^2\ a0^3}$$

In[50]:=	equ
Out[50]=	$${-3\ a^3\ a0^4\ x^2 + 3\ a3\ x^2 - 2\ a^4\ a0^5\ x^3 - 2\ a\ a0\ a3\ x^3}$$
	$+ 4\ a4\ x^3 - a^5\ a0^6\ x^4 - 2\ a^2\ a0^2\ a3\ x^4 - 2\ a\ a0\ a4\ x^4$
	$- 2\ a^3\ a0^3\ a3\ x^5 - 2\ a^2\ a0^2\ a4\ x^5 - a\ a3^2\ x^6 - 2\ a^3\ a0^3\ a4\ x^6$
	$- 2\ a\ a3\ a4\ x^7 - a\ a4^2\ x^8$
In[51]:=	Coefficient[equ, x^2]
Out[51]=	$${-3\ a^3\ a0^4 + 3\ a3}$$
In[52]:=	Solve[%==0, a3]
Out[52]=	$${\{\{a3\mathrel{-\negthinspace\negthinspace>}a^3\ a0^4\}\} }$$
In[53]:=	a3 = a3/.%[[1]]
Out[53]=	$${a^3\ a0^4}$$

In[54]:=	equ
Out[54]=	$${-4\ a^4\ a0^5\ x^3 + 4\ a4\ x^3 - 3\ a^5\ a0^6\ x^4 - 2\ a\ a0\ a4\ x^4}$$
	$- 2\ a^6\ a0^7\ x^5 - 2\ a^2\ a0^2\ a4\ x^5 - a^7\ a0^8\ x^6 - 2\ a^3\ a0^3\ a4\ x^6$
	$- 2\ a^4\ a0^4\ a4\ x^7 - a\ a4^2\ x^8$
In[55]:=	Coefficient[equ, x^3]
Out[55]=	$${-4\ a^4\ a0^5 + 4\ a4}$$
In[56]:=	Solve[%==0, a4]
Out[56]=	$${\{\{a4\mathrel{-\negthinspace\negthinspace>}a^4\ a0^5\}\} }$$
In[57]:=	a4 = a4/.%[[1]]
Out[57]=	$${a^4\ a0^5}$$
In[58]:=	y
Out[58]=	$${a0 + a\ a0^2\ x + a^2\ a0^3\ x^2 + a^3\ a0^4\ x^3 + a^4\ a0^5\ x^4}$$
In[59]:=	Factor[%]
Out[59]=	$${a0\ (1 + a\ a0\ x + a^2\ a0^2\ x^2 + a^3\ a0^3\ x^3 + a^4\ a0^4\ x^4)}$$

This looks like the first few terms of the series for

$$y(x) = {a_0\over 1 - a_0 a x}$$ (5.2)

which is indeed the solution of the differential equation (5.1. So we see that in this sort of bookkeeping exercise the use of computer algebra can be of considerable assistance.