Internal problem ID [8095]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 515.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {f \left (y^{2}+x^{2}\right ) \left (\left (y^{\prime }\right )^{2}+1\right )-\left (x y^{\prime }-y\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.344 (sec). Leaf size: 154
dsolve(f(y(x)^2+x^2)*(diff(y(x),x)^2+1)-(x*diff(y(x),x)-y(x))^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -i x \\ y \relax (x ) = i x \\ y \relax (x ) = \RootOf \left (x^{2}+\textit {\_Z}^{2}-f \left (\textit {\_Z}^{2}+x^{2}\right )\right ) \\ y \relax (x ) = \frac {x}{\tan \left (\RootOf \left (-2 \textit {\_Z} -\left (\int _{}^{\frac {x^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )+1\right )}{\tan \left (\textit {\_Z} \right )^{2}}}\frac {\sqrt {-f \left (\textit {\_a} \right )^{2}+f \left (\textit {\_a} \right ) \textit {\_a}}}{\textit {\_a} \left (f \left (\textit {\_a} \right )-\textit {\_a} \right )}d \textit {\_a} \right )+2 c_{1}\right )\right )} \\ y \relax (x ) = \frac {x}{\tan \left (\RootOf \left (-2 \textit {\_Z} +\int _{}^{\frac {x^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )+1\right )}{\tan \left (\textit {\_Z} \right )^{2}}}\frac {\sqrt {-f \left (\textit {\_a} \right )^{2}+f \left (\textit {\_a} \right ) \textit {\_a}}}{\textit {\_a} \left (f \left (\textit {\_a} \right )-\textit {\_a} \right )}d \textit {\_a} +2 c_{1}\right )\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.238 (sec). Leaf size: 1922
DSolve[-(-y[x] + x*y'[x])^2 + f[x^2 + y[x]^2]*(1 + y'[x]^2)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\int _1^x\left (\frac {\sqrt {f\left (K[1]^2+y(x)^2\right ) \left (K[1]^2+y(x)^2-f\left (K[1]^2+y(x)^2\right )\right )} K[1]}{f\left (K[1]^2+y(x)^2\right ) \left (K[1]^2+y(x)^2\right )}-\frac {\sqrt {f\left (K[1]^2+y(x)^2\right ) \left (K[1]^2+y(x)^2-f\left (K[1]^2+y(x)^2\right )\right )} K[1]}{f\left (K[1]^2+y(x)^2\right ) \left (K[1]^2+y(x)^2-f\left (K[1]^2+y(x)^2\right )\right )}+\frac {y(x)}{K[1]^2+y(x)^2}\right )dK[1]+\int _1^{y(x)}\left (-\frac {x}{x^2+K[2]^2}-\int _1^x\left (-\frac {2 K[2]^2}{\left (K[1]^2+K[2]^2\right )^2}-\frac {2 K[1] \sqrt {f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )} f'\left (K[1]^2+K[2]^2\right ) K[2]}{f\left (K[1]^2+K[2]^2\right )^2 \left (K[1]^2+K[2]^2\right )}+\frac {2 K[1] \sqrt {f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )} f'\left (K[1]^2+K[2]^2\right ) K[2]}{f\left (K[1]^2+K[2]^2\right )^2 \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )}-\frac {2 K[1] \sqrt {f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )} K[2]}{f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2\right )^2}+\frac {K[1] \sqrt {f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )} \left (2 K[2]-2 K[2] f'\left (K[1]^2+K[2]^2\right )\right )}{f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )^2}+\frac {K[1] \left (2 K[2] \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right ) f'\left (K[1]^2+K[2]^2\right )+f\left (K[1]^2+K[2]^2\right ) \left (2 K[2]-2 K[2] f'\left (K[1]^2+K[2]^2\right )\right )\right )}{2 f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2\right ) \sqrt {f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )}}-\frac {K[1] \left (2 K[2] \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right ) f'\left (K[1]^2+K[2]^2\right )+f\left (K[1]^2+K[2]^2\right ) \left (2 K[2]-2 K[2] f'\left (K[1]^2+K[2]^2\right )\right )\right )}{2 f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right ) \sqrt {f\left (K[1]^2+K[2]^2\right ) \left (K[1]^2+K[2]^2-f\left (K[1]^2+K[2]^2\right )\right )}}+\frac {1}{K[1]^2+K[2]^2}\right )dK[1]+\frac {K[2] \sqrt {f\left (x^2+K[2]^2\right ) \left (x^2+K[2]^2-f\left (x^2+K[2]^2\right )\right )}}{f\left (x^2+K[2]^2\right ) \left (x^2+K[2]^2\right )}-\frac {K[2] \sqrt {f\left (x^2+K[2]^2\right ) \left (x^2+K[2]^2-f\left (x^2+K[2]^2\right )\right )}}{f\left (x^2+K[2]^2\right ) \left (x^2+K[2]^2-f\left (x^2+K[2]^2\right )\right )}\right )dK[2]=c_1,y(x)\right ] \\ \text {Solve}\left [\int _1^x\left (-\frac {\sqrt {f\left (K[3]^2+y(x)^2\right ) \left (K[3]^2+y(x)^2-f\left (K[3]^2+y(x)^2\right )\right )} K[3]}{f\left (K[3]^2+y(x)^2\right ) \left (K[3]^2+y(x)^2\right )}+\frac {\sqrt {f\left (K[3]^2+y(x)^2\right ) \left (K[3]^2+y(x)^2-f\left (K[3]^2+y(x)^2\right )\right )} K[3]}{f\left (K[3]^2+y(x)^2\right ) \left (K[3]^2+y(x)^2-f\left (K[3]^2+y(x)^2\right )\right )}+\frac {y(x)}{K[3]^2+y(x)^2}\right )dK[3]+\int _1^{y(x)}\left (-\frac {x}{x^2+K[4]^2}-\int _1^x\left (-\frac {2 K[4]^2}{\left (K[3]^2+K[4]^2\right )^2}+\frac {2 K[3] \sqrt {f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )} f'\left (K[3]^2+K[4]^2\right ) K[4]}{f\left (K[3]^2+K[4]^2\right )^2 \left (K[3]^2+K[4]^2\right )}-\frac {2 K[3] \sqrt {f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )} f'\left (K[3]^2+K[4]^2\right ) K[4]}{f\left (K[3]^2+K[4]^2\right )^2 \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )}+\frac {2 K[3] \sqrt {f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )} K[4]}{f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2\right )^2}-\frac {K[3] \sqrt {f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )} \left (2 K[4]-2 K[4] f'\left (K[3]^2+K[4]^2\right )\right )}{f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )^2}-\frac {K[3] \left (2 K[4] \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right ) f'\left (K[3]^2+K[4]^2\right )+f\left (K[3]^2+K[4]^2\right ) \left (2 K[4]-2 K[4] f'\left (K[3]^2+K[4]^2\right )\right )\right )}{2 f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2\right ) \sqrt {f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )}}+\frac {K[3] \left (2 K[4] \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right ) f'\left (K[3]^2+K[4]^2\right )+f\left (K[3]^2+K[4]^2\right ) \left (2 K[4]-2 K[4] f'\left (K[3]^2+K[4]^2\right )\right )\right )}{2 f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right ) \sqrt {f\left (K[3]^2+K[4]^2\right ) \left (K[3]^2+K[4]^2-f\left (K[3]^2+K[4]^2\right )\right )}}+\frac {1}{K[3]^2+K[4]^2}\right )dK[3]-\frac {K[4] \sqrt {f\left (x^2+K[4]^2\right ) \left (x^2+K[4]^2-f\left (x^2+K[4]^2\right )\right )}}{f\left (x^2+K[4]^2\right ) \left (x^2+K[4]^2\right )}+\frac {K[4] \sqrt {f\left (x^2+K[4]^2\right ) \left (x^2+K[4]^2-f\left (x^2+K[4]^2\right )\right )}}{f\left (x^2+K[4]^2\right ) \left (x^2+K[4]^2-f\left (x^2+K[4]^2\right )\right )}\right )dK[4]=c_1,y(x)\right ] \\ \end{align*}