58.14.7 problem 7

Internal problem ID [14847]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.1. Exercises page 232
Problem number : 7
Date solved : Thursday, October 02, 2025 at 09:55:44 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 44

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

\[ y = \left (1-\frac {1}{6} x^{3}+\frac {3}{40} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {3}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 49

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

\[ y(x)\to c_1 \left (\frac {3 x^5}{40}-\frac {x^3}{6}+1\right )+c_2 \left (\frac {3 x^5}{40}-\frac {x^4}{12}-\frac {x^3}{6}+x\right ) \]

✓ Sympy. Time used: 0.285 (sec). Leaf size: 42

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

\[ y{\left (x \right )} = C_{2} \left (\frac {3 x^{5}}{40} - \frac {x^{3}}{6} + 1\right ) + C_{1} x \left (\frac {3 x^{4}}{40} - \frac {x^{3}}{12} - \frac {x^{2}}{6} + 1\right ) + O\left (x^{6}\right ) \]