85.7.2 problem 1 (b)

Internal problem ID [22456]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 1. Differential equations in general. A Exercises at page 32
Problem number : 1 (b)
Date solved : Thursday, October 02, 2025 at 08:39:47 PM
CAS classification : [`y=_G(x,y')`]

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

With initial conditions

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

✓ Maple. Time used: 0.134 (sec). Leaf size: 56

ode:=diff(y(x),x) = 1/(x^2+y(x)^2); 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \operatorname {RootOf}\left (-\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {\textit {\_Z}^{2}}{2}\right ) \operatorname {BesselY}\left (-\frac {3}{4}, \frac {1}{2}\right ) \textit {\_Z} -\operatorname {BesselJ}\left (\frac {1}{4}, \frac {\textit {\_Z}^{2}}{2}\right ) \operatorname {BesselY}\left (-\frac {3}{4}, \frac {1}{2}\right ) x +\operatorname {BesselY}\left (-\frac {3}{4}, \frac {\textit {\_Z}^{2}}{2}\right ) \operatorname {BesselJ}\left (-\frac {3}{4}, \frac {1}{2}\right ) \textit {\_Z} +\operatorname {BesselY}\left (\frac {1}{4}, \frac {\textit {\_Z}^{2}}{2}\right ) \operatorname {BesselJ}\left (-\frac {3}{4}, \frac {1}{2}\right ) x \right ) \]

✗ Mathematica

ode=D[y[x],{x,1}]==1/(x^2+y[x]^2); 
ic={y[0]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

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

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