23.1.585 problem 578

Internal problem ID [5192]
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 : 578
Date solved : Tuesday, September 30, 2025 at 11:52:01 AM
CAS classification : [[_homogeneous, `class D`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 33

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

\begin{align*} y &= \sqrt {{\mathrm e}^{\frac {1}{x}} c_1 +x^{2}} \\ y &= -\sqrt {{\mathrm e}^{\frac {1}{x}} c_1 +x^{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.204 (sec). Leaf size: 195

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

\begin{align*} y(x)&\to -e^{\left .\frac {1}{2}\right /x} \sqrt {2 \int _1^x\frac {1}{2} e^{-\frac {1}{K[1]}} (2 K[1]+1)dK[1]+c_1}\\ y(x)&\to e^{\left .\frac {1}{2}\right /x} \sqrt {2 \int _1^x\frac {1}{2} e^{-\frac {1}{K[1]}} (2 K[1]+1)dK[1]+c_1}\\ y(x)&\to -\sqrt {2} e^{\left .\frac {1}{2}\right /x} \sqrt {\int _1^x\frac {1}{2} e^{-\frac {1}{K[1]}} (2 K[1]+1)dK[1]}\\ y(x)&\to \sqrt {2} e^{\left .\frac {1}{2}\right /x} \sqrt {\int _1^x\frac {1}{2} e^{-\frac {1}{K[1]}} (2 K[1]+1)dK[1]} \end{align*}

✓ Sympy. Time used: 0.319 (sec). Leaf size: 32

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

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