23.1.602 problem 595

Internal problem ID [5209]
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 : 595
Date solved : Tuesday, September 30, 2025 at 11:55:47 AM
CAS classification : [_separable]

\begin{align*} \left (1+y\right ) y^{\prime } \sqrt {x^{2}+1}&=y^{3} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 57

ode:=(1+y(x))*diff(y(x),x)*(x^2+1)^(1/2) = y(x)^3; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.262 (sec). Leaf size: 68

ode=(1+y[x])*D[y[x],x]*Sqrt[1+x^2]==y[x]^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.424 (sec). Leaf size: 53

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(sqrt(x**2 + 1)*(y(x) + 1)*Derivative(y(x), x) - y(x)**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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