6.11.14 problem 25

Internal problem ID [1853]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.1 Exercises. Page 318
Problem number : 25
Date solved : Tuesday, September 30, 2025 at 05:20:33 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 30

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

\[ y = \frac {c_1 \left (1+\operatorname {O}\left (x^{6}\right )\right ) x +c_2 \left (1-3 x^{2}-\frac {1}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 25

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

\[ y(x)\to c_1 \left (-\frac {x^2}{2}+\frac {1}{x^2}-3\right )+\frac {c_2}{x} \]

✓ Sympy. Time used: 0.477 (sec). Leaf size: 14

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

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