Internal problem ID [10802]
Book: Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS,
MOSCOW, Third printing 1977.
Section: Chapter 1, First-Order Differential Equations. Problems page 88
Problem number: Problem 53.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
\[ \boxed {y \left ({y^{\prime }}^{2}+1\right )-a=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 1583
dsolve(y(x)*(1+diff(y(x),x)^2)=a,y(x), singsol=all)
\begin{align*} y \left (x \right ) = a \\ y \left (x \right ) = \frac {\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+\textit {\_Z}^{2} a^{2}+4 c_{1} \textit {\_Z} a -4 \textit {\_Z} a x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) \operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+\textit {\_Z}^{2} a^{2}+4 c_{1} \textit {\_Z} a -4 \textit {\_Z} a x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right ) a}{2}+\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+\textit {\_Z}^{2} a^{2}+4 c_{1} \textit {\_Z} a -4 \textit {\_Z} a x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) c_{1} -\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+\textit {\_Z}^{2} a^{2}+4 c_{1} \textit {\_Z} a -4 \textit {\_Z} a x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) x +\frac {a}{2} \\ y \left (x \right ) = \frac {\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+\textit {\_Z}^{2} a^{2}+4 c_{1} \textit {\_Z} a -4 \textit {\_Z} a x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) \left (-\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+\textit {\_Z}^{2} a^{2}+4 c_{1} \textit {\_Z} a -4 \textit {\_Z} a x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right ) a -4 c_{1} +4 x \right )}{2}+\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+\textit {\_Z}^{2} a^{2}+4 c_{1} \textit {\_Z} a -4 \textit {\_Z} a x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) c_{1} -\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}+4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+\textit {\_Z}^{2} a^{2}+4 c_{1} \textit {\_Z} a -4 \textit {\_Z} a x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) x +\frac {a}{2} \\ y \left (x \right ) = \frac {\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+\textit {\_Z}^{2} a^{2}-4 c_{1} \textit {\_Z} a +4 \textit {\_Z} a x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) \operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+\textit {\_Z}^{2} a^{2}-4 c_{1} \textit {\_Z} a +4 \textit {\_Z} a x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right ) a}{2}-\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+\textit {\_Z}^{2} a^{2}-4 c_{1} \textit {\_Z} a +4 \textit {\_Z} a x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) c_{1} +\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+\textit {\_Z}^{2} a^{2}-4 c_{1} \textit {\_Z} a +4 \textit {\_Z} a x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) x +\frac {a}{2} \\ y \left (x \right ) = \frac {\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+\textit {\_Z}^{2} a^{2}-4 c_{1} \textit {\_Z} a +4 \textit {\_Z} a x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) \left (-\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+\textit {\_Z}^{2} a^{2}-4 c_{1} \textit {\_Z} a +4 \textit {\_Z} a x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right ) a +4 c_{1} -4 x \right )}{2}-\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+\textit {\_Z}^{2} a^{2}-4 c_{1} \textit {\_Z} a +4 \textit {\_Z} a x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) c_{1} +\tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} a^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} a \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} a x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{1}^{2}-8 \tan \left (\textit {\_Z} \right )^{2} c_{1} x +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+\textit {\_Z}^{2} a^{2}-4 c_{1} \textit {\_Z} a +4 \textit {\_Z} a x +4 c_{1}^{2}-8 c_{1} x -a^{2}+4 x^{2}\right )\right ) x +\frac {a}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.476 (sec). Leaf size: 106
DSolve[y[x]*(1+y'[x]^2)==a,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [a \arctan \left (\frac {\sqrt {\text {$\#$1}}}{\sqrt {a-\text {$\#$1}}}\right )-\sqrt {\text {$\#$1}} \sqrt {a-\text {$\#$1}}\&\right ][-x+c_1] \\ y(x)\to \text {InverseFunction}\left [a \arctan \left (\frac {\sqrt {\text {$\#$1}}}{\sqrt {a-\text {$\#$1}}}\right )-\sqrt {\text {$\#$1}} \sqrt {a-\text {$\#$1}}\&\right ][x+c_1] \\ y(x)\to a \\ \end{align*}