20.9.13 problem Problem 13

Internal problem ID [3757]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 8, Linear differential equations of order n. Section 8.7, The Variation of Parameters Method. page 556
Problem number : Problem 13
Date solved : Sunday, March 30, 2025 at 02:07:30 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 26

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

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

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 37

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

\[ y(x)\to \frac {1}{2} e^{m x} \left (2 x \arctan (x)-\log \left (x^2+1\right )+2 (c_2 x+c_1)\right ) \]

✗ Sympy

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

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