23.5.6 problem 3(e)

Internal problem ID [4183]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 7. Special functions. Exercises at page 124
Problem number : 3(e)
Date solved : Sunday, March 30, 2025 at 02:41:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\frac {\left (x^{2}+4 x +2\right ) \left (\left (1-x \right ) y^{\prime }+y\right )}{x \left (-x^{2}+2\right )}&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 44

Order:=6; 
ode:=diff(diff(y(x),x),x)-(x^2+4*x+2)/x/(-x^2+2)*((1-x)*diff(y(x),x)+y(x)) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{2} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (-2+2 x +4 x^{2}+4 x^{3}+2 x^{4}+\frac {2}{3} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 65

ode=D[y[x],{x,2}]-(x^2+4*x+2)/(x*(2-x^2))*( (1-x)*D[y[x],x]+y[x] )==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},{y[x]},{x,0,5}]
 

\[ \{y(x)\}\to c_1 \left (-\frac {5 x^4}{4}-\frac {5 x^3}{2}-\frac {5 x^2}{2}-x+1\right )+c_2 \left (\frac {x^6}{24}+\frac {x^5}{6}+\frac {x^4}{2}+x^3+x^2\right ) \]

✓ Sympy. Time used: 1.151 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) - ((1 - x)*Derivative(y(x), x) + y(x))*(x**2 + 4*x + 2)/(x*(2 - x**2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{2} + C_{1} + O\left (x^{6}\right ) \]