Internal problem ID [8846]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 511.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
\[ \boxed {\left (a^{2} \sqrt {x^{2}+y^{2}}-x^{2}\right ) {y^{\prime }}^{2}+2 x y^{\prime } y+a^{2} \sqrt {x^{2}+y^{2}}-y^{2}=0} \]
✓ Solution by Maple
Time used: 7.843 (sec). Leaf size: 311
dsolve((a^2*(y(x)^2+x^2)^(1/2)-x^2)*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)+a^2*(y(x)^2+x^2)^(1/2)-y(x)^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -i x \\ y \left (x \right ) &= i x \\ \frac {2 \sqrt {-a^{2}+\sqrt {y \left (x \right )^{2}+x^{2}}}\, \sqrt {a^{2} \left (y \left (x \right )^{2}+x^{2}\right )^{2} \left (-a^{2}+\sqrt {y \left (x \right )^{2}+x^{2}}\right )}\, \arctan \left (\frac {\sqrt {-a^{2}+\sqrt {y \left (x \right )^{2}+x^{2}}}}{a}\right )-a \left (y \left (x \right )^{2}+x^{2}\right ) \left (a^{2}-\sqrt {y \left (x \right )^{2}+x^{2}}\right ) \left (c_{1} -\arctan \left (\frac {x}{y \left (x \right )}\right )\right )}{a \left (y \left (x \right )^{2}+x^{2}\right ) \left (a^{2}-\sqrt {y \left (x \right )^{2}+x^{2}}\right )} &= 0 \\ \frac {-2 \sqrt {-a^{2}+\sqrt {y \left (x \right )^{2}+x^{2}}}\, \sqrt {a^{2} \left (y \left (x \right )^{2}+x^{2}\right )^{2} \left (-a^{2}+\sqrt {y \left (x \right )^{2}+x^{2}}\right )}\, \arctan \left (\frac {\sqrt {-a^{2}+\sqrt {y \left (x \right )^{2}+x^{2}}}}{a}\right )-a \left (y \left (x \right )^{2}+x^{2}\right ) \left (a^{2}-\sqrt {y \left (x \right )^{2}+x^{2}}\right ) \left (c_{1} -\arctan \left (\frac {x}{y \left (x \right )}\right )\right )}{a \left (y \left (x \right )^{2}+x^{2}\right ) \left (a^{2}-\sqrt {y \left (x \right )^{2}+x^{2}}\right )} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 42.919 (sec). Leaf size: 229
DSolve[-y[x]^2 + a^2*Sqrt[x^2 + y[x]^2] + 2*x*y[x]*y'[x] + (-x^2 + a^2*Sqrt[x^2 + y[x]^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^2 \left (x^2+y(x)^2\right ) \left (\sqrt {x^2+y(x)^2}-a^2\right )} \arctan \left (\frac {\sqrt {\sqrt {x^2+y(x)^2}-a^2}}{a}\right )}{a \sqrt {x^2+y(x)^2} \sqrt {\sqrt {x^2+y(x)^2}-a^2}}&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {2 \sqrt {a^2 \left (x^2+y(x)^2\right ) \left (\sqrt {x^2+y(x)^2}-a^2\right )} \arctan \left (\frac {\sqrt {\sqrt {x^2+y(x)^2}-a^2}}{a}\right )}{a \sqrt {x^2+y(x)^2} \sqrt {\sqrt {x^2+y(x)^2}-a^2}}+\arctan \left (\frac {x}{y(x)}\right )&=c_1,y(x)\right ] \\ \end{align*}