69.20.31 problem 670

Internal problem ID [18452]
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 : 670
Date solved : Friday, October 03, 2025 at 07:32:36 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

With initial conditions

\begin{align*} y \left (-\infty \right )&=0 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.076 (sec). Leaf size: 12

ode:=(1-x)*diff(diff(y(x),x),x)+x*diff(y(x),x)-y(x) = (x-1)^2*exp(x); 
ic:=[y(-infinity) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✗ Mathematica

ode=(1-x)*D[y[x],{x,2}]+x*D[y[x],x]-y[x]==(x-1)^2*Exp[x]; 
ic={y[-Infinity]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

{}

✗ Sympy

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