Internal problem ID [3904]
Book: Differential Equations, By George Boole F.R.S. 1865
Section: Chapter 7
Problem number: 7.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {y-a y^{\prime }-\sqrt {1+\left (y^{\prime }\right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.138 (sec). Leaf size: 77
dsolve(y(x)=a*diff(y(x),x)+sqrt(1+(diff(y(x),x))^2),y(x), singsol=all)
\begin{align*} x -\left (\int _{}^{y \relax (x )}-\frac {\left (a -1\right ) \left (a +1\right )}{-a \textit {\_a} +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} \right )-c_{1} = 0 \\ x -\left (\int _{}^{y \relax (x )}\frac {\left (a -1\right ) \left (a +1\right )}{a \textit {\_a} +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} \right )-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.776 (sec). Leaf size: 278
DSolve[y[x]==a*y'[x]+Sqrt[1+(y'[x])^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {2 \log \left (\sqrt {\text {$\#$1}^2+a^2-1}+\text {$\#$1}\right )-a \log \left (-a \sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}-a^2+1\right )+a \log \left (-a \sqrt {\text {$\#$1}^2+a^2-1}+\text {$\#$1}-a^2+1\right )-a \log \left (1-\text {$\#$1}^2\right )+a \log (1-\text {$\#$1})-a \log (\text {$\#$1}+1)}{2-2 a^2}\&\right ]\left [\frac {x}{a^2-1}+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\frac {2 \log \left (\sqrt {\text {$\#$1}^2+a^2-1}+\text {$\#$1}\right )-a \log \left (-a \sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}-a^2+1\right )+a \log \left (-a \sqrt {\text {$\#$1}^2+a^2-1}+\text {$\#$1}-a^2+1\right )+a \log \left (1-\text {$\#$1}^2\right )+a \log (1-\text {$\#$1})-a \log (\text {$\#$1}+1)}{2 \left (a^2-1\right )}\&\right ]\left [\frac {x}{a^2-1}+c_1\right ] \\ y(x)\to 1 \\ \end{align*}