6.9.10 problem 10

Internal problem ID [1766]
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 : 10
Date solved : Tuesday, September 30, 2025 at 05:19:10 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{2} y^{\prime \prime }+2 x \left (x -1\right ) y^{\prime }+\left (x^{2}-2 x +2\right ) y&=x^{3} {\mathrm e}^{2 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: 0.007 (sec). Leaf size: 23

ode:=x^2*diff(diff(y(x),x),x)+2*x*(x-1)*diff(y(x),x)+(x^2-2*x+2)*y(x) = x^3*exp(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x \left ({\mathrm e}^{3 x}+9 c_1 x +9 c_2 \right ) {\mathrm e}^{-x}}{9} \]

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 30

ode=x^2*D[y[x],{x,2}]+2*x*(x-1)*D[y[x],x]+(x^2-2*x+2)*y[x]==x^3*Exp[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{9} e^{-x} x \left (e^{3 x}+9 (c_2 x+c_1)\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**3*exp(2*x) + x**2*Derivative(y(x), (x, 2)) + 2*x*(x - 1)*Derivative(y(x), x) + (x**2 - 2*x + 2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (x**3*exp(2*x) - x**2*y(x) - x**2*Derivative(y(x), (x, 2)) + 2*x*y(x) - 2*y(x))/(2*x*(x - 1)) cannot be solved by the factorable group method