Internal problem ID [7640]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 59.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {y^{\prime }-a \sqrt {y^{2}+1}-b=0} \end {gather*}
✓ Solution by Maple
Time used: 0.001 (sec). Leaf size: 26
dsolve(diff(y(x),x) - a*sqrt(y(x)^2+1) - b=0,y(x), singsol=all)
\[ x -\left (\int _{}^{y \relax (x )}\frac {1}{a \sqrt {\textit {\_a}^{2}+1}+b}d \textit {\_a} \right )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.568 (sec). Leaf size: 123
DSolve[y'[x] - a*Sqrt[y[x]^2+1] - b==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {\frac {2 b \text {ArcTan}\left (\frac {\left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right ) a+b}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\tanh ^{-1}\left (\frac {\text {$\#$1}}{\sqrt {\text {$\#$1}^2+1}}\right )}{a}\&\right ][x+c_1] \\ y(x)\to -\frac {\sqrt {b^2-a^2}}{a} \\ y(x)\to \frac {\sqrt {b^2-a^2}}{a} \\ \end{align*}