50.18.5 problem 1(e)

Internal problem ID [8099]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.3. Second-Order Linear Equations: Ordinary Points. Page 169
Problem number : 1(e)
Date solved : Wednesday, March 05, 2025 at 05:29:50 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*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 49

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

\[ y = \left (1-\frac {1}{2} x^{2}+\frac {5}{24} x^{4}-\frac {17}{144} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{6} x^{5}-\frac {13}{126} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 56

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

\[ y(x)\to c_2 \left (-\frac {13 x^7}{126}+\frac {x^5}{6}-\frac {x^3}{3}+x\right )+c_1 \left (-\frac {17 x^6}{144}+\frac {5 x^4}{24}-\frac {x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.871 (sec). Leaf size: 42

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 = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {17 x^{6}}{144} + \frac {5 x^{4}}{24} - \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {x^{4}}{6} - \frac {x^{2}}{3} + 1\right ) + O\left (x^{8}\right ) \]