74.15.51 problem 53 (d)

Internal problem ID [16419]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.7, page 195
Problem number : 53 (d)
Date solved : Monday, March 31, 2025 at 02:52:50 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} \left (x^{2}+1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}+1\right ) y^{\prime }+4 y&=\arctan \left (x \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 37

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

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

✓ Mathematica. Time used: 1.124 (sec). Leaf size: 208

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

\[ y(x)\to \frac {\exp \left (\int _1^x\frac {K[1]+2 i}{K[1]^2+1}dK[1]\right ) \left (\int _1^x-\frac {\exp \left (\int _1^{K[3]}\frac {K[1]+2 i}{K[1]^2+1}dK[1]\right ) \arctan (K[3]) \int _1^{K[3]}\exp \left (-2 \int _1^{K[2]}\frac {K[1]+2 i}{K[1]^2+1}dK[1]\right )dK[2]}{\left (K[3]^2+1\right )^{3/2}}dK[3]+\int _1^x\exp \left (-2 \int _1^{K[2]}\frac {K[1]+2 i}{K[1]^2+1}dK[1]\right )dK[2] \left (\int _1^x\frac {\exp \left (\int _1^{K[4]}\frac {K[1]+2 i}{K[1]^2+1}dK[1]\right ) \arctan (K[4])}{\left (K[4]^2+1\right )^{3/2}}dK[4]+c_2\right )+c_1\right )}{\sqrt {x^2+1}} \]

✗ Sympy

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

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