Internal problem ID [8096]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 516.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A]]
Solve \begin {gather*} \boxed {\left (y^{2}+x^{2}\right ) f \left (\frac {x}{\sqrt {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: 2.953 (sec). Leaf size: 70
dsolve((y(x)^2+x^2)*f(x/(y(x)^2+x^2)^(1/2))*(diff(y(x),x)^2+1)-(x*diff(y(x),x)-y(x))^2=0,y(x), singsol=all)
\[ y \relax (x ) = \RootOf \left (-\ln \relax (x )+\int _{}^{\textit {\_Z}}\frac {-\textit {\_a} f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )+\sqrt {-f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )^{2}+f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )}}{\left (\textit {\_a}^{2}+1\right ) f \left (\frac {1}{\sqrt {\textit {\_a}^{2}+1}}\right )}d \textit {\_a} +c_{1}\right ) x \]
✓ Solution by Mathematica
Time used: 2.018 (sec). Leaf size: 253
DSolve[-(-y[x] + x*y'[x])^2 + f[x/Sqrt[x^2 + y[x]^2]]*(x^2 + y[x]^2)*(1 + y'[x]^2)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {f\left (\frac {1}{\sqrt {K[1]^2+1}}\right ) K[1]^2+f\left (\frac {1}{\sqrt {K[1]^2+1}}\right )-1}{\sqrt {f\left (\frac {1}{\sqrt {K[1]^2+1}}\right )} (K[1]-i) (K[1]+i) \left (\sqrt {f\left (\frac {1}{\sqrt {K[1]^2+1}}\right )} K[1]+i \sqrt {f\left (\frac {1}{\sqrt {K[1]^2+1}}\right )-1}\right )}dK[1]=-\log (x)+c_1,y(x)\right ] \\ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {f\left (\frac {1}{\sqrt {K[2]^2+1}}\right ) K[2]^2+f\left (\frac {1}{\sqrt {K[2]^2+1}}\right )-1}{\sqrt {f\left (\frac {1}{\sqrt {K[2]^2+1}}\right )} (K[2]-i) (K[2]+i) \left (\sqrt {f\left (\frac {1}{\sqrt {K[2]^2+1}}\right )} K[2]-i \sqrt {f\left (\frac {1}{\sqrt {K[2]^2+1}}\right )-1}\right )}dK[2]=-\log (x)+c_1,y(x)\right ] \\ \end{align*}