Internal problem ID [3914]
Book: Differential Equations, By George Boole F.R.S. 1865
Section: Chapter 7
Problem number: 17.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational, _dAlembert]
Solve \begin {gather*} \boxed {x +y y^{\prime }-a \sqrt {\left (y^{\prime }\right )^{2}+1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.422 (sec). Leaf size: 349
dsolve(x+y(x)*diff(y(x),x)=a*sqrt(1+(diff(y(x),x))^2),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {a \sqrt {\tan ^{2}\left (\RootOf \left (a^{2} \textit {\_Z}^{2} \cos \left (2 \textit {\_Z} \right )+4 \sin \left (\textit {\_Z} \right ) a x \textit {\_Z} +2 c_{1} \textit {\_Z} a \cos \left (2 \textit {\_Z} \right )-a^{2} \textit {\_Z}^{2}+4 c_{1} x \sin \left (\textit {\_Z} \right )+c_{1}^{2} \cos \left (2 \textit {\_Z} \right )+a^{2} \cos \left (2 \textit {\_Z} \right )-2 c_{1} \textit {\_Z} a -c_{1}^{2}+a^{2}-2 x^{2}\right )\right )+1}-x}{\tan \left (\RootOf \left (a^{2} \textit {\_Z}^{2} \cos \left (2 \textit {\_Z} \right )+4 \sin \left (\textit {\_Z} \right ) a x \textit {\_Z} +2 c_{1} \textit {\_Z} a \cos \left (2 \textit {\_Z} \right )-a^{2} \textit {\_Z}^{2}+4 c_{1} x \sin \left (\textit {\_Z} \right )+c_{1}^{2} \cos \left (2 \textit {\_Z} \right )+a^{2} \cos \left (2 \textit {\_Z} \right )-2 c_{1} \textit {\_Z} a -c_{1}^{2}+a^{2}-2 x^{2}\right )\right )} \\ y \relax (x ) = \frac {a \sqrt {\tan ^{2}\left (\RootOf \left (a^{2} \textit {\_Z}^{2} \cos \left (2 \textit {\_Z} \right )-4 \sin \left (\textit {\_Z} \right ) a x \textit {\_Z} +2 c_{1} \textit {\_Z} a \cos \left (2 \textit {\_Z} \right )-a^{2} \textit {\_Z}^{2}-4 c_{1} x \sin \left (\textit {\_Z} \right )+c_{1}^{2} \cos \left (2 \textit {\_Z} \right )+a^{2} \cos \left (2 \textit {\_Z} \right )-2 c_{1} \textit {\_Z} a -c_{1}^{2}+a^{2}-2 x^{2}\right )\right )+1}-x}{\tan \left (\RootOf \left (a^{2} \textit {\_Z}^{2} \cos \left (2 \textit {\_Z} \right )-4 \sin \left (\textit {\_Z} \right ) a x \textit {\_Z} +2 c_{1} \textit {\_Z} a \cos \left (2 \textit {\_Z} \right )-a^{2} \textit {\_Z}^{2}-4 c_{1} x \sin \left (\textit {\_Z} \right )+c_{1}^{2} \cos \left (2 \textit {\_Z} \right )+a^{2} \cos \left (2 \textit {\_Z} \right )-2 c_{1} \textit {\_Z} a -c_{1}^{2}+a^{2}-2 x^{2}\right )\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 3.059 (sec). Leaf size: 319
DSolve[x+y[x]*y'[x]==a*Sqrt[1+(y'[x])^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {a^2 \left (-\text {ArcTan}\left (\frac {\sqrt {a^2} x-\sqrt {a^2 \left (-a^2+x^2+y(x)^2\right )}}{a^2-a y(x)}\right )\right )-a^2 \text {ArcTan}\left (\frac {\sqrt {a^2} x-\sqrt {a^2 \left (-a^2+x^2+y(x)^2\right )}}{a^2+a y(x)}\right )+\sqrt {a^2} a \text {ArcTan}\left (\frac {\sqrt {a^2} x}{a y(x)}\right )-\sqrt {a^2 \left (-a^2+x^2+y(x)^2\right )}}{a^2}=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {a^2 \text {ArcTan}\left (\frac {\sqrt {a^2} x-\sqrt {a^2 \left (-a^2+x^2+y(x)^2\right )}}{a^2-a y(x)}\right )+a^2 \text {ArcTan}\left (\frac {\sqrt {a^2} x-\sqrt {a^2 \left (-a^2+x^2+y(x)^2\right )}}{a^2+a y(x)}\right )+\sqrt {a^2} a \text {ArcTan}\left (\frac {\sqrt {a^2} x}{a y(x)}\right )+\sqrt {a^2 \left (-a^2+x^2+y(x)^2\right )}}{a^2}=c_1,y(x)\right ] \\ \end{align*}