Internal problem ID [12150]
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]
\[ \boxed {y \left ({y^{\prime }}^{2}+1\right )=a} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 339
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 {\left (\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) a +\textit {\_Z} a +2 c_{1} -2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) a +\textit {\_Z} a +2 c_{1} -2 x \right )\right ) a -2 x +2 c_{1} \right ) \tan \left (\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) a +\textit {\_Z} a +2 c_{1} -2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) a +\textit {\_Z} a +2 c_{1} -2 x \right )\right )\right )}{2}+\frac {a}{2} \\ y \left (x \right ) &= \frac {\left (-\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) a +\textit {\_Z} a +2 c_{1} -2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) a +\textit {\_Z} a +2 c_{1} -2 x \right )\right ) a +2 x -2 c_{1} \right ) \tan \left (\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) a +\textit {\_Z} a +2 c_{1} -2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) a +\textit {\_Z} a +2 c_{1} -2 x \right )\right )\right )}{2}+\frac {a}{2} \\ y \left (x \right ) &= \frac {\left (\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) a -\textit {\_Z} a +2 c_{1} -2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) a -\textit {\_Z} a +2 c_{1} -2 x \right )\right ) a +2 x -2 c_{1} \right ) \tan \left (\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) a -\textit {\_Z} a +2 c_{1} -2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) a -\textit {\_Z} a +2 c_{1} -2 x \right )\right )\right )}{2}+\frac {a}{2} \\ y \left (x \right ) &= \frac {\left (-\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) a -\textit {\_Z} a +2 c_{1} -2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) a -\textit {\_Z} a +2 c_{1} -2 x \right )\right ) a -2 x +2 c_{1} \right ) \tan \left (\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) a -\textit {\_Z} a +2 c_{1} -2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) a -\textit {\_Z} a +2 c_{1} -2 x \right )\right )\right )}{2}+\frac {a}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.661 (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*}