23.2.127 problem 129

Internal problem ID [5482]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 129
Date solved : Tuesday, September 30, 2025 at 12:45:14 PM
CAS classification : [[_homogeneous, `class C`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 53

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

\begin{align*} y &= 0 \\ y &= \frac {\left (x +1+\sqrt {\left (1+x \right ) \left (1+c_1 \right )}\right )^{2}}{1+x} \\ y &= \frac {\left (-x -1+\sqrt {\left (1+x \right ) \left (1+c_1 \right )}\right )^{2}}{1+x} \\ \end{align*}

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 57

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

\begin{align*} y(x)&\to x-c_1 \sqrt {x+1}+1+\frac {c_1{}^2}{4}\\ y(x)&\to x+c_1 \sqrt {x+1}+1+\frac {c_1{}^2}{4}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.576 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = \frac {C_{1}^{2}}{4} - C_{1} \sqrt {x + 1} + x + 1 \]