23.1.604 problem 598

Internal problem ID [5211]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 598
Date solved : Tuesday, September 30, 2025 at 11:55:52 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 22

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

\[ y = -\tan \left (\operatorname {RootOf}\left (x^{2}+2 \tan \left (\textit {\_Z} \right )+2 c_1 -2 \textit {\_Z} \right )\right ) \]

✓ Mathematica. Time used: 0.14 (sec). Leaf size: 52

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {K[1]^2}{K[1]^2+1}dK[1]\&\right ]\left [\frac {x^2}{2}+c_1\right ]\\ y(x)&\to -i\\ y(x)&\to i \end{align*}

✓ Sympy. Time used: 0.159 (sec). Leaf size: 14

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

\[ - \frac {x^{2}}{2} + y{\left (x \right )} - \operatorname {atan}{\left (y{\left (x \right )} \right )} = C_{1} \]