58.14.13 problem 13

Internal problem ID [14853]
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 : 13
Date solved : Thursday, October 02, 2025 at 09:55:48 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 18

Order:=6; 
ode:=(x^2+1)*diff(diff(y(x),x),x)+x*diff(y(x),x)+2*x*y(x) = 0; 
ic:=[y(0) = 2, D(y)(0) = 3]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

\[ y = 2+3 x -\frac {7}{6} x^{3}-\frac {1}{2} x^{4}+\frac {21}{40} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 29

ode=(x^2+1)*D[y[x],{x,2}]+x*D[y[x],x]+2*x*y[x]==0; 
ic={y[0]==2,Derivative[1][y][0] ==3}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {21 x^5}{40}-\frac {x^4}{2}-\frac {7 x^3}{6}+3 x+2 \]

✓ Sympy. Time used: 0.294 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*y(x) + x*Derivative(y(x), x) + (x**2 + 1)*Derivative(y(x), (x, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(x), x), x, 0): 3} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {3 x^{5}}{20} - \frac {x^{3}}{3} + 1\right ) + C_{1} x \left (\frac {3 x^{4}}{40} - \frac {x^{3}}{6} - \frac {x^{2}}{6} + 1\right ) + O\left (x^{6}\right ) \]