Internal problem ID [3913]
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]
\[ \boxed {x +y y^{\prime }-a \sqrt {1+{y^{\prime }}^{2}}=0} \]
✓ Solution by Maple
Time used: 0.11 (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 \left (x \right ) = \frac {\sqrt {{\tan \left (\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2} \cos \left (2 \textit {\_Z} \right )+2 c_{1} \textit {\_Z} a \cos \left (2 \textit {\_Z} \right )+4 \sin \left (\textit {\_Z} \right ) a x \textit {\_Z} -a^{2} \textit {\_Z}^{2}+c_{1}^{2} \cos \left (2 \textit {\_Z} \right )+a^{2} \cos \left (2 \textit {\_Z} \right )+4 c_{1} x \sin \left (\textit {\_Z} \right )-2 c_{1} \textit {\_Z} a -c_{1}^{2}+a^{2}-2 x^{2}\right )\right )}^{2}+1}\, a -x}{\tan \left (\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2} \cos \left (2 \textit {\_Z} \right )+2 c_{1} \textit {\_Z} a \cos \left (2 \textit {\_Z} \right )+4 \sin \left (\textit {\_Z} \right ) a x \textit {\_Z} -a^{2} \textit {\_Z}^{2}+c_{1}^{2} \cos \left (2 \textit {\_Z} \right )+a^{2} \cos \left (2 \textit {\_Z} \right )+4 c_{1} x \sin \left (\textit {\_Z} \right )-2 c_{1} \textit {\_Z} a -c_{1}^{2}+a^{2}-2 x^{2}\right )\right )} \\ y \left (x \right ) = \frac {a \sqrt {{\tan \left (\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2} \cos \left (2 \textit {\_Z} \right )+2 c_{1} \textit {\_Z} a \cos \left (2 \textit {\_Z} \right )-4 \sin \left (\textit {\_Z} \right ) a x \textit {\_Z} -a^{2} \textit {\_Z}^{2}+c_{1}^{2} \cos \left (2 \textit {\_Z} \right )+a^{2} \cos \left (2 \textit {\_Z} \right )-4 c_{1} x \sin \left (\textit {\_Z} \right )-2 c_{1} \textit {\_Z} a -c_{1}^{2}+a^{2}-2 x^{2}\right )\right )}^{2}+1}-x}{\tan \left (\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2} \cos \left (2 \textit {\_Z} \right )+2 c_{1} \textit {\_Z} a \cos \left (2 \textit {\_Z} \right )-4 \sin \left (\textit {\_Z} \right ) a x \textit {\_Z} -a^{2} \textit {\_Z}^{2}+c_{1}^{2} \cos \left (2 \textit {\_Z} \right )+a^{2} \cos \left (2 \textit {\_Z} \right )-4 c_{1} x \sin \left (\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: 3.521 (sec). Leaf size: 388
DSolve[x+y[x]*y'[x]==a*Sqrt[1+(y'[x])^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {\frac {2 a \sqrt {a^2 y(x)^2-a^4} \arctan \left (\frac {a x \sqrt {y(x)^2-a^2}}{y(x) \left (\sqrt {a^2 \left (y(x)^2-a^2\right )}-\sqrt {a^2 \left (-a^2+x^2+y(x)^2\right )}\right )+a^2 x}\right )}{\sqrt {y(x)^2-a^2}}-\sqrt {a^2 \left (-a^2+x^2+y(x)^2\right )}}{a^2}-\frac {a \sqrt {y(x)^2-a^2} \arctan \left (\frac {\sqrt {y(x)^2-a^2}}{a}\right )}{\sqrt {a^2 \left (y(x)^2-a^2\right )}}=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {a \sqrt {y(x)^2-a^2} \arctan \left (\frac {\sqrt {y(x)^2-a^2}}{a}\right )}{\sqrt {a^2 \left (y(x)^2-a^2\right )}}+\frac {\sqrt {a^2 \left (-a^2+x^2+y(x)^2\right )}-\frac {2 a \sqrt {a^2 y(x)^2-a^4} \arctan \left (\frac {a x \sqrt {y(x)^2-a^2}}{y(x) \left (\sqrt {a^2 \left (-a^2+x^2+y(x)^2\right )}-\sqrt {a^2 \left (y(x)^2-a^2\right )}\right )+a^2 x}\right )}{\sqrt {y(x)^2-a^2}}}{a^2}=c_1,y(x)\right ] \\ \end{align*}