6.10.26 problem 26

Internal problem ID [1830]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.7 Variation of Parameters. Page 262
Problem number : 26
Date solved : Tuesday, September 30, 2025 at 05:20:11 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{2} y^{\prime \prime }-2 x \left (x +2\right ) y^{\prime }+\left (x^{2}+4 x +6\right ) y&=2 x \,{\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 18

ode:=x^2*diff(diff(y(x),x),x)-2*x*(x+2)*diff(y(x),x)+(x^2+4*x+6)*y(x) = 2*x*exp(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{x} x \left (c_1 \,x^{2}+c_2 x +1\right ) \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 22

ode=x^2*D[y[x],{x,2}]-2*x*(x+2)*D[y[x],x]+(x^2+4*x+6)*y[x]==2*x*Exp[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - 2*x*(x + 2)*Derivative(y(x), x) - 2*x*exp(x) + (x**2 + 4*x + 6)*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*Derivative(y(x), (x, 2)) + 4*x*y(x) - 2*x*exp(x) + 6*y(x))/(2*x*(x + 2)) cannot be solved by the factorable group method