Internal problem ID [8149]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 569.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_Clairaut]
Solve \begin {gather*} \boxed {\left (\left (y^{\prime }\right )^{2}+1\right ) \left (\sin ^{2}\left (x y^{\prime }-y\right )\right )-1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.459 (sec). Leaf size: 147
dsolve((diff(y(x),x)^2+1)*sin(x*diff(y(x),x)-y(x))^2-1=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\sqrt {1-x}\, \sqrt {\frac {1}{x}}\, x -\arcsin \left (\sqrt {\frac {1}{x}}\, x \right ) \\ y \relax (x ) = \sqrt {1-x}\, \sqrt {\frac {1}{x}}\, x +\arcsin \left (\sqrt {\frac {1}{x}}\, x \right ) \\ y \relax (x ) = -\sqrt {x +1}\, \sqrt {-\frac {1}{x}}\, x -\arcsin \left (\sqrt {-\frac {1}{x}}\, x \right ) \\ y \relax (x ) = \sqrt {x +1}\, \sqrt {-\frac {1}{x}}\, x +\arcsin \left (\sqrt {-\frac {1}{x}}\, x \right ) \\ y \relax (x ) = c_{1} x -\arcsin \left (\frac {1}{\sqrt {c_{1}^{2}+1}}\right ) \\ y \relax (x ) = c_{1} x +\arcsin \left (\frac {1}{\sqrt {c_{1}^{2}+1}}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.353 (sec). Leaf size: 71
DSolve[-1 + Sin[y[x] - x*y'[x]]^2*(1 + y'[x]^2)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 x-\frac {1}{2} \text {ArcCos}\left (1-\frac {2}{1+c_1{}^2}\right ) \\ y(x)\to \frac {1}{2} \text {ArcCos}\left (1-\frac {2}{1+c_1{}^2}\right )+c_1 x \\ y(x)\to -\frac {\pi }{2} \\ y(x)\to \frac {\pi }{2} \\ \end{align*}