21.2.4 problem 4

Internal problem ID [4304]
Book : An introduction to the solution and applications of differential equations, J.W. Searl, 1966
Section : Chapter 4, Ex. 4.2
Problem number : 4
Date solved : Tuesday, September 30, 2025 at 07:16:48 AM
CAS classification : [_separable]

\begin{align*} \sqrt {x^{2}+1}\, y^{\prime }+\sqrt {1+y^{2}}&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 11

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

\[ y = -\sinh \left (\operatorname {arcsinh}\left (x \right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.121 (sec). Leaf size: 29

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

\begin{align*} y(x)&\to -\sinh (\text {arcsinh}(x)-c_1)\\ y(x)&\to -i\\ y(x)&\to i \end{align*}

✓ Sympy. Time used: 0.191 (sec). Leaf size: 8

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

\[ y{\left (x \right )} = \sinh {\left (C_{1} - \operatorname {asinh}{\left (x \right )} \right )} \]