20.9.17 problem Problem 17

Internal problem ID [3761]
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 17
Date solved : Sunday, March 30, 2025 at 02:07:37 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=\frac {4 \,{\mathrm e}^{-2 x}}{x^{2}+1}+2 x^{2}-1 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 39

ode:=diff(diff(y(x),x),x)+4*diff(y(x),x)+4*y(x) = 4*exp(-2*x)/(x^2+1)+2*x^2-1; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (x -1\right )^{2}}{2}+{\mathrm e}^{-2 x} \left (c_1 x +4 \arctan \left (x \right ) x +c_2 -4 \ln \left (2\right )-2 \ln \left (x^{2}+1\right )\right ) \]

✓ Mathematica. Time used: 0.706 (sec). Leaf size: 59

ode=D[y[x],{x,2}]+4*D[y[x],x]+4*y[x]==4*Exp[-2*x]/(1+x^2)+2*x^2-1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{2} e^{-2 x} \left (8 x \arctan (x)+e^{2 x} x^2-4 \log \left (x^2+1\right )-2 e^{2 x} x+e^{2 x}+2 c_2 x+2 c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x**2 + 4*y(x) + 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)) + 1 - 4*exp(-2*x)/(x**2 + 1),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (2*x**4*exp(2*x) - 4*x**2*y(x)*exp(2*x) - x**2*exp(2*x)*Derivative(y(x), (x, 2)) + x**2*exp(2*x) - 4*y(x)*exp(2*x) - exp(2*x)*Derivative(y(x), (x, 2)) - exp(2*x) + 4)*exp(-2*x)/(4*(x**2 + 1)) cannot be solved by the factorable group method