23.1.407 problem 394

Internal problem ID [5014]
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 : 394
Date solved : Tuesday, September 30, 2025 at 11:23:01 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 39

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

\[ c_1 +x^{2}+x \sqrt {x^{2}+1}+\operatorname {arcsinh}\left (x \right )+y \sqrt {1+y^{2}}+\operatorname {arcsinh}\left (y\right )-y^{2} = 0 \]

✓ Mathematica. Time used: 0.598 (sec). Leaf size: 84

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\frac {1}{2} \left (\text {$\#$1} \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )-\log \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )\right )\&\right ]\left [\frac {1}{2} \left (\log \left (\sqrt {x^2+1}-x\right )-x \left (\sqrt {x^2+1}+x\right )\right )+c_1\right ] \end{align*}

✓ Sympy. Time used: 1.141 (sec). Leaf size: 151

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

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