69.20.18 problem 657

Internal problem ID [18439]
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 : 657
Date solved : Thursday, October 02, 2025 at 03:11:56 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y-2 y^{\prime }+y^{\prime \prime }&=\frac {{\mathrm e}^{x}}{x^{2}+1} \end{align*}

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

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

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

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 48

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

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

✗ Sympy

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

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