6.9.11 problem 11

Internal problem ID [1767]
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 : 11
Date solved : Tuesday, September 30, 2025 at 05:19:11 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=x \,{\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 24

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

\[ y = \frac {{\mathrm e}^{x} \left (3 c_1 \,x^{2}+x^{3}+3 c_2 \right )}{3 x} \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 32

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

\begin{align*} y(x)&\to \frac {e^x \left (2 x^3+3 c_2 x^2+6 c_1\right )}{6 x} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (x**2*y(x) - x**2*exp(x) + x**2*Derivative(y(x), (x, 2)) - x*y(x) - y(x))/(x*(2*x - 1)) cannot be solved by the factorable group method