23.1.519 problem 509

Internal problem ID [5126]
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 : 509
Date solved : Tuesday, September 30, 2025 at 11:43:28 AM
CAS classification : [[_homogeneous, `class D`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 27

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

\begin{align*} y &= \sqrt {x \left (c_1 x -2\right )} \\ y &= -\sqrt {x \left (c_1 x -2\right )} \\ \end{align*}

✓ Mathematica. Time used: 0.188 (sec). Leaf size: 42

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

\begin{align*} y(x)&\to -\sqrt {x} \sqrt {-2+c_1 x}\\ y(x)&\to \sqrt {x} \sqrt {-2+c_1 x} \end{align*}

✓ Sympy. Time used: 0.263 (sec). Leaf size: 26

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

\[ \left [ y{\left (x \right )} = - \sqrt {x \left (C_{1} x - 2\right )}, \ y{\left (x \right )} = \sqrt {x \left (C_{1} x - 2\right )}\right ] \]