6.9.28 problem 28

Internal problem ID [1784]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page 253
Problem number : 28
Date solved : Tuesday, September 30, 2025 at 05:19:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=\frac {1}{x} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 16

ode:=(2*x+1)*x*diff(diff(y(x),x),x)-2*(2*x^2-1)*diff(y(x),x)-4*(1+x)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_1}{x}+c_2 \,{\mathrm e}^{2 x} \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 28

ode=(2*x+1)*x*D[y[x],{x,2}]-2*(2*x^2-1)*D[y[x],x]-4*(x+1)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {c_2 e^{2 x+1} x+c_1}{\sqrt {e} x} \end{align*}

✗ Sympy

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

False