Internal problem ID [8092]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 512.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (a \left (x^{2}+y^{2}\right )^{\frac {3}{2}}-x^{2}\right ) \left (y^{\prime }\right )^{2}+2 x y^{\prime } y+a \left (x^{2}+y^{2}\right )^{\frac {3}{2}}-y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 7.81 (sec). Leaf size: 149
dsolve((a*(y(x)^2+x^2)^(3/2)-x^2)*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)+a*(y(x)^2+x^2)^(3/2)-y(x)^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -i x \\ y \relax (x ) = i x \\ y \relax (x ) = \frac {x}{\tan \left (\RootOf \left (-\textit {\_Z} +\int _{}^{\frac {x^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )+1\right )}{\tan \left (\textit {\_Z} \right )^{2}}}-\frac {\sqrt {-\textit {\_a}^{\frac {5}{2}} a \left (a \sqrt {\textit {\_a}}-1\right )}\, \left (a \sqrt {\textit {\_a}}+1\right )}{2 \textit {\_a}^{2} \left (\textit {\_a} \,a^{2}-1\right )}d \textit {\_a} +c_{1}\right )\right )} \\ y \relax (x ) = \frac {x}{\tan \left (\RootOf \left (-\textit {\_Z} +\int _{}^{\frac {x^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )+1\right )}{\tan \left (\textit {\_Z} \right )^{2}}}\frac {\sqrt {-\textit {\_a}^{\frac {5}{2}} a \left (a \sqrt {\textit {\_a}}-1\right )}\, \left (a \sqrt {\textit {\_a}}+1\right )}{2 \textit {\_a}^{2} \left (\textit {\_a} \,a^{2}-1\right )}d \textit {\_a} +c_{1}\right )\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 70.324 (sec). Leaf size: 305
DSolve[-y[x]^2 + a*(x^2 + y[x]^2)^(3/2) + 2*x*y[x]*y'[x] + (-x^2 + a*(x^2 + y[x]^2)^(3/2))*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\text {ArcTan}\left (\frac {x}{y(x)}\right )-\frac {2 \sqrt {a \left (x^2+y(x)^2\right )^2 \left (-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}\right )} \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {x^2+y(x)^2}}{\sqrt {-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}}}\right )}{\sqrt {a} \left (x^2+y(x)^2\right ) \sqrt {-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}}}=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {2 \sqrt {a \left (x^2+y(x)^2\right )^2 \left (-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}\right )} \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {x^2+y(x)^2}}{\sqrt {-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}}}\right )}{\sqrt {a} \left (x^2+y(x)^2\right ) \sqrt {-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}}}+\text {ArcTan}\left (\frac {x}{y(x)}\right )=c_1,y(x)\right ] \\ \end{align*}