44.11.17 problem 5(b)

Internal problem ID [9287]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Second-Order Linear Equations. Section 2.3. THE METHOD OF VARIATION OF PARAMETERS. Page 71
Problem number : 5(b)
Date solved : Tuesday, September 30, 2025 at 06:16:08 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 23

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

\[ y = \frac {c_2}{x}+{\mathrm e}^{x} c_1 -\frac {x^{2}}{3}-x -1 \]

✓ Mathematica. Time used: 0.181 (sec). Leaf size: 268

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

\begin{align*} y(x)&\to \exp \left (\int _1^x-\frac {K[1]+2}{2 K[1]+2}dK[1]-\frac {1}{2} \int _1^x\frac {2-K[2]^2}{K[2]^2+K[2]}dK[2]\right ) \left (\int _1^x-\exp \left (\int _1^{K[4]}-\frac {K[1]+2}{2 K[1]+2}dK[1]+\frac {1}{2} \int _1^{K[4]}\frac {2-K[2]^2}{K[2]^2+K[2]}dK[2]\right ) (K[4]+1) \int _1^{K[4]}\exp \left (-2 \int _1^{K[3]}-\frac {K[1]+2}{2 K[1]+2}dK[1]\right )dK[3]dK[4]+\int _1^x\exp \left (-2 \int _1^{K[3]}-\frac {K[1]+2}{2 K[1]+2}dK[1]\right )dK[3] \left (\int _1^x\exp \left (\int _1^{K[5]}-\frac {K[1]+2}{2 K[1]+2}dK[1]+\frac {1}{2} \int _1^{K[5]}\frac {2-K[2]^2}{K[2]^2+K[2]}dK[2]\right ) (K[5]+1)dK[5]+c_2\right )+c_1\right ) \end{align*}

✗ Sympy

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

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