Internal problem ID [8407]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 827.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {-y+x^{3} \sqrt {y^{2}+x^{2}}-x^{2} \sqrt {y^{2}+x^{2}}\, y}{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 49
dsolve(diff(y(x),x) = -(-y(x)+x^3*(y(x)^2+x^2)^(1/2)-x^2*(y(x)^2+x^2)^(1/2)*y(x))/x,y(x), singsol=all)
\[ \ln \left (\frac {2 x \left (\sqrt {2 x^{2}+2 y \relax (x )^{2}}+y \relax (x )+x \right )}{y \relax (x )-x}\right )+\frac {\sqrt {2}\, x^{3}}{3}-\ln \relax (x )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 1.351 (sec). Leaf size: 71
DSolve[y'[x] == (y[x] - x^3*Sqrt[x^2 + y[x]^2] + x^2*y[x]*Sqrt[x^2 + y[x]^2])/x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} x \left (\sqrt {2} \tanh \left (\frac {x^3+3 c_1}{3 \sqrt {2}}\right )-\frac {1}{1+\sqrt {2} \tanh \left (\frac {x^3+3 c_1}{3 \sqrt {2}}\right )}+1\right ) \\ y(x)\to x \\ \end{align*}