68.15.53 problem 53 (f)

Internal problem ID [17779]
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 (f)
Date solved : Thursday, October 02, 2025 at 02:28:07 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 26

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); 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {x^{2} \arctan \left (x \right )+\arctan \left (x \right )+3 x}{4 x^{2}+4} \]

✓ Mathematica. Time used: 0.417 (sec). Leaf size: 420

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={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} 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\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]-\int _1^0\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]+\exp \left (\int _1^0\frac {K[1]+2 i}{K[1]^2+1}dK[1]\right )\right )+\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 _1^0-\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]-\exp \left (\int _1^0\frac {K[1]+2 i}{K[1]^2+1}dK[1]\right ) \int _1^0\exp \left (-2 \int _1^{K[2]}\frac {K[1]+2 i}{K[1]^2+1}dK[1]\right )dK[2]\right )}{\sqrt {x^2+1}} \end{align*}

✗ 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 = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 1} 
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