23.1.248 problem 243

Internal problem ID [4855]
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 : 243
Date solved : Tuesday, September 30, 2025 at 08:45:10 AM
CAS classification : [_rational, _Bernoulli]

\begin{align*} 2 x y^{\prime }&=\left (1+x -6 y^{2}\right ) y \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 54

ode:=2*x*diff(y(x),x) = (1+x-6*y(x)^2)*y(x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\sqrt {\left (c_1 \,{\mathrm e}^{-x}+6\right ) x}}{c_1 \,{\mathrm e}^{-x}+6} \\ y &= -\frac {\sqrt {\left (c_1 \,{\mathrm e}^{-x}+6\right ) x}}{c_1 \,{\mathrm e}^{-x}+6} \\ \end{align*}

✓ Mathematica. Time used: 0.522 (sec). Leaf size: 73

ode=2*x*D[y[x],x]==(1+x-6*y[x]^2)*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {e^{\frac {x+1}{2}} \sqrt {x}}{\sqrt {6 e^{x+1}+c_1}}\\ y(x)&\to \frac {e^{\frac {x+1}{2}} \sqrt {x}}{\sqrt {6 e^{x+1}+c_1}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.497 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), x) - (x - 6*y(x)**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \sqrt {\frac {x e^{x}}{C_{1} + 6 e^{x}}}, \ y{\left (x \right )} = \sqrt {\frac {x e^{x}}{C_{1} + 6 e^{x}}}\right ] \]