Internal problem ID [4422]
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], _dAlembert]
\[ \boxed {y y^{\prime }-a \sqrt {1+{y^{\prime }}^{2}}=-x} \]
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 237
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 ) &= \csc \left (\operatorname {RootOf}\left (\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} -\cos \left (\textit {\_Z} \right ) a -x \right ) \left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} +\cos \left (\textit {\_Z} \right ) a -x \right )\right )\right ) \operatorname {csgn}\left (\sec \left (\operatorname {RootOf}\left (\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} -\cos \left (\textit {\_Z} \right ) a -x \right ) \left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} +\cos \left (\textit {\_Z} \right ) a -x \right )\right )\right )\right ) a -\cot \left (\operatorname {RootOf}\left (\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} -\cos \left (\textit {\_Z} \right ) a -x \right ) \left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} +\cos \left (\textit {\_Z} \right ) a -x \right )\right )\right ) x \\ y \left (x \right ) &= \csc \left (\operatorname {RootOf}\left (\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} +\cos \left (\textit {\_Z} \right ) a +x \right ) \left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} -\cos \left (\textit {\_Z} \right ) a +x \right )\right )\right ) a \,\operatorname {csgn}\left (\sec \left (\operatorname {RootOf}\left (\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} +\cos \left (\textit {\_Z} \right ) a +x \right ) \left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} -\cos \left (\textit {\_Z} \right ) a +x \right )\right )\right )\right )-\cot \left (\operatorname {RootOf}\left (\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} +\cos \left (\textit {\_Z} \right ) a +x \right ) \left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\sin \left (\textit {\_Z} \right ) c_{1} -\cos \left (\textit {\_Z} \right ) a +x \right )\right )\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 3.538 (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*}