20.9.4 problem Problem 4

Internal problem ID [3748]
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 4
Date solved : Sunday, March 30, 2025 at 02:07:14 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+9 y&=\frac {2 \,{\mathrm e}^{-3 x}}{x^{2}+1} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 27

ode:=diff(diff(y(x),x),x)+6*diff(y(x),x)+9*y(x) = 2*exp(-3*x)/(x^2+1); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-3 x} \left (2 \arctan \left (x \right ) x +c_1 x -\ln \left (x^{2}+1\right )+c_2 \right ) \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 31

ode=D[y[x],{x,2}]+6*D[y[x],x]+9*y[x]==2*Exp[-3*x]/(x^2+1); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-3 x} \left (2 x \arctan (x)-\log \left (x^2+1\right )+c_2 x+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*y(x) + 6*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 2*exp(-3*x)/(x**2 + 1),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-9*x**2*y(x)*exp(3*x) - x**2*exp(3*x)*Derivative(y(x), (x, 2)) - 9*y(x)*exp(3*x) - exp(3*x)*Derivative(y(x), (x, 2)) + 2)*exp(-3*x)/(6*(x**2 + 1)) cannot be solved by the factorable group method