6.12.11 problem 11

Internal problem ID [1865]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.2 SERIES SOLUTIONS NEAR AN ORDINARY POINT I. Exercises 7.2. Page 329
Problem number : 11
Date solved : Tuesday, September 30, 2025 at 05:20:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 34

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

\[ y(x)\to -\frac {x^5}{6}+\frac {5 x^4}{12}+\frac {x^3}{3}-x^2-x+2 \]

✓ Sympy. Time used: 0.256 (sec). Leaf size: 31

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

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