6.12.6 problem 6

Internal problem ID [1860]
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 : 6
Date solved : Tuesday, September 30, 2025 at 05:20:38 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }+\frac {y}{4}&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 39

Order:=6; 
ode:=(x^2+1)*diff(diff(y(x),x),x)+2*x*diff(y(x),x)+1/4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{8} x^{2}+\frac {25}{384} x^{4}\right ) y \left (0\right )+\left (x -\frac {3}{8} x^{3}+\frac {147}{640} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 42

ode=(1+x^2)*D[y[x],{x,2}]+2*x*D[y[x],x]+1/4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {147 x^5}{640}-\frac {3 x^3}{8}+x\right )+c_1 \left (\frac {25 x^4}{384}-\frac {x^2}{8}+1\right ) \]

✓ Sympy. Time used: 0.270 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), x) + (x**2 + 1)*Derivative(y(x), (x, 2)) + y(x)/4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {25 x^{4}}{384} - \frac {x^{2}}{8} + 1\right ) + C_{1} x \left (1 - \frac {3 x^{2}}{8}\right ) + O\left (x^{6}\right ) \]