Internal problem ID [3905]
Book: Differential Equations, By George Boole F.R.S. 1865
Section: Chapter 7
Problem number: 8.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {x -a y^{\prime }-\sqrt {\left (y^{\prime }\right )^{2}+1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.234 (sec). Leaf size: 195
dsolve(x=a*diff(y(x),x)+sqrt(1+(diff(y(x),x))^2),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {x \sqrt {a^{2}+x^{2}-1}}{2 \left (a -1\right ) \left (a +1\right )}+\frac {\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right ) a^{2}}{2 \left (a -1\right ) \left (a +1\right )}-\frac {\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right )}{2 \left (a -1\right ) \left (a +1\right )}+\frac {a \,x^{2}}{2 \left (a -1\right ) \left (a +1\right )}+c_{1} \\ y \relax (x ) = -\frac {x \sqrt {a^{2}+x^{2}-1}}{2 \left (a -1\right ) \left (a +1\right )}-\frac {\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right ) a^{2}}{2 \left (a -1\right ) \left (a +1\right )}+\frac {\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right )}{2 \left (a -1\right ) \left (a +1\right )}+\frac {a \,x^{2}}{2 \left (a -1\right ) \left (a +1\right )}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.037 (sec). Leaf size: 113
DSolve[x==a*y'[x]+Sqrt[1+(y'[x])^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \left (\frac {x \left (\sqrt {a^2+x^2-1}+a x\right )}{a^2-1}-\log \left (\sqrt {a^2+x^2-1}-x\right )\right )+c_1 \\ y(x)\to \frac {1}{2} \left (\frac {x \left (a x-\sqrt {a^2+x^2-1}\right )}{a^2-1}+\log \left (\sqrt {a^2+x^2-1}-x\right )\right )+c_1 \\ \end{align*}