75.20.9 problem 644

Internal problem ID [17072]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.5 Linear equations with variable coefficients. The Lagrange method. Exercises page 148
Problem number : 644
Date solved : Monday, March 31, 2025 at 03:39:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x -1\right ) y^{\prime \prime }-x y^{\prime }+y&=\left (x -1\right )^{2} {\mathrm e}^{x} \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: 22

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

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

✓ Mathematica. Time used: 0.234 (sec). Leaf size: 248

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*Derivative(y(x), x) - (x - 1)**2*exp(x) + (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*exp(x) + 2*exp(x) + Derivative(y(x), (x, 2))) + y(x) - exp(x) - Derivative(y(x), (x, 2)))/x cannot be solved by the factorable group method