48.9.12 problem 12

Internal problem ID [9968]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number : 12
Date solved : Tuesday, September 30, 2025 at 06:45:21 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 38

Order:=8; 
ode:=x^2*diff(diff(y(x),x),x)-x*(x^2+1)*diff(y(x),x)+(-x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = x \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\frac {1}{48} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\left (-\frac {1}{4} x^{2}-\frac {3}{32} x^{4}-\frac {11}{576} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 86

ode=x^2*D[y[x],{x,2}]-x*(1+x^2)*D[y[x],x]+(1-x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 x \left (\frac {x^6}{48}+\frac {x^4}{8}+\frac {x^2}{2}+1\right )+c_2 \left (x \left (-\frac {11 x^6}{576}-\frac {3 x^4}{32}-\frac {x^2}{4}\right )+x \left (\frac {x^6}{48}+\frac {x^4}{8}+\frac {x^2}{2}+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.340 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - x*(x**2 + 1)*Derivative(y(x), x) + (1 - x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} x \left (\frac {x^{6}}{48} + \frac {x^{4}}{8} + \frac {x^{2}}{2} + 1\right ) + O\left (x^{8}\right ) \]