67.9.30 problem 14.3 (f)

Internal problem ID [16578]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 14. Higher order equations and the reduction of order method. Additional exercises page 277
Problem number : 14.3 (f)
Date solved : Thursday, October 02, 2025 at 01:36:35 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 18

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

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

✓ Mathematica. Time used: 0.138 (sec). Leaf size: 241

ode=(x+1)*D[y[x],{x,2}]+x*D[y[x],x]-y[x]==(1+x)^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 {K[2]}{K[2]+1}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 {K[2]}{K[2]+1}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 {K[2]}{K[2]+1}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*Derivative(y(x), x) - (x + 1)**2 + (x + 1)*Derivative(y(x), (x, 2)) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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