Internal problem ID [2634]
Book: An introduction to the solution and applications of differential equations, J.W. Searl,
1966
Section: Chapter 4, Ex. 4.2
Problem number: 4.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {\sqrt {x^{2}+1}\, y^{\prime }+\sqrt {1+y^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 11
dsolve(sqrt(1+x^2)*diff(y(x),x)+sqrt(1+y(x)^2)=0,y(x), singsol=all)
\[ y \relax (x ) = -\sinh \left (\arcsinh \relax (x )+c_{1}\right ) \]
✓ Solution by Mathematica
Time used: 15.091 (sec). Leaf size: 110
DSolve[Sqrt[1+x^2]*y'[x]+Sqrt[1+y[x]^2]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\tanh \left (\tanh ^{-1}\left (\frac {x}{\sqrt {x^2+1}}\right )-c_1\right )}{\sqrt {\text {sech}^2\left (\tanh ^{-1}\left (\frac {x}{\sqrt {x^2+1}}\right )-c_1\right )}} \\ y(x)\to \frac {\tanh \left (\tanh ^{-1}\left (\frac {x}{\sqrt {x^2+1}}\right )-c_1\right )}{\sqrt {\text {sech}^2\left (\tanh ^{-1}\left (\frac {x}{\sqrt {x^2+1}}\right )-c_1\right )}} \\ y(x)\to -i \\ y(x)\to i \\ \end{align*}