23.1.105 problem 103

Internal problem ID [4712]
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 : 103
Date solved : Tuesday, September 30, 2025 at 08:18:31 AM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }+2 y \left (1-x \sqrt {y}\right )&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 17

ode:=diff(y(x),x)+2*y(x)*(1-x*y(x)^(1/2)) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {1}{\sqrt {y}}-x -1-{\mathrm e}^{x} c_1 = 0 \]

✓ Mathematica. Time used: 0.101 (sec). Leaf size: 41

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

\begin{align*} y(x)&\to \frac {4 e^{-2 x}}{\left (\int _1^x2 e^{-K[1]} K[1]dK[1]-2 c_1\right ){}^2}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.237 (sec). Leaf size: 36

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

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