23.1.730 problem 728

Internal problem ID [5337]
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 : 728
Date solved : Tuesday, September 30, 2025 at 12:30:48 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 19

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

\[ -y-2 \sqrt {x +y}-c_1 = 0 \]

✓ Mathematica. Time used: 0.054 (sec). Leaf size: 39

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

\begin{align*} y(x)&\to -2 \sqrt {x+1+c_1}+2+c_1\\ y(x)&\to 2 \sqrt {x+1+c_1}+2+c_1 \end{align*}

✓ Sympy. Time used: 0.729 (sec). Leaf size: 31

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

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