58.12.21 problem 21

Internal problem ID [14806]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.4. Variation of parameters. Exercises page 162
Problem number : 21
Date solved : Thursday, October 02, 2025 at 09:55:07 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

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

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

\[ y = \ln \left (x \right ) x^{2}+\left (c_2 -1\right ) x^{2}+\left (c_1 -2\right ) x +c_1 \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 31

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

\begin{align*} y(x)&\to x^2 \log (x)+(-1+c_1) x^2-(2+c_2) x-c_2 \end{align*}

✗ Sympy

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

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