Internal problem ID [8090]
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]]
\[ \boxed {\left (a \left (x^{2}+y^{2}\right )^{\frac {3}{2}}-x^{2}\right ) {y^{\prime }}^{2}+2 x y^{\prime } y+a \left (x^{2}+y^{2}\right )^{\frac {3}{2}}-y^{2}=0} \]
✓ Solution by Maple
Time used: 8.5 (sec). Leaf size: 127
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 \left (x \right ) = -i x \\ y \left (x \right ) = i x \\ y \left (x \right ) = \frac {x}{\tan \left (\operatorname {RootOf}\left (-4 \textit {\_Z} -\left (\int _{}^{\frac {x^{2} \left (\tan \left (\textit {\_Z} \right )^{2}+1\right )}{\tan \left (\textit {\_Z} \right )^{2}}}\frac {2 \sqrt {-a \,\textit {\_a}^{\frac {17}{2}} \left (a \sqrt {\textit {\_a}}-1\right ) \left (4 a \cos \left (2\right ) \sqrt {\textit {\_a}}+4 \textit {\_a} \,a^{2}-4 a \sqrt {\textit {\_a}}+\cos \left (2\right )^{2}-2 \cos \left (2\right )+1\right )}}{\left (a \cos \left (2\right ) \sqrt {\textit {\_a}}+2 \textit {\_a} \,a^{2}-3 a \sqrt {\textit {\_a}}-\cos \left (2\right )+1\right ) \textit {\_a}^{5}}d \textit {\_a} \right )+4 c_{1} \right )\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 50.021 (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 [\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 )} \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 )} \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}}}+\arctan \left (\frac {x}{y(x)}\right )=c_1,y(x)\right ] \\ \end{align*}