73.17.44 problem 44

Internal problem ID [15507]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 25. Review exercises for part III. page 447
Problem number : 44
Date solved : Monday, March 31, 2025 at 01:39:53 PM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 33

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

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

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 63

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

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

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**4*Derivative(y(x), (x, 2)) + x**2*y(x) - x**2*Derivative(y(x), (x, 2)) + y(x) + 1)/(x**3 + x) cannot be solved by the factorable group method