Internal problem ID [9214]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 879.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\[ \boxed {y^{\prime }+\frac {-x y-y+x^{2} \sqrt {x^{2}+y^{2}}-x \sqrt {x^{2}+y^{2}}\, y}{x \left (1+x \right )}=0} \]
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 55
dsolve(diff(y(x),x) = -(-x*y(x)-y(x)+(y(x)^2+x^2)^(1/2)*x^2-x*(y(x)^2+x^2)^(1/2)*y(x))/x/(x+1),y(x), singsol=all)
\[ \ln \left (\frac {2 x \left (\sqrt {2 y \left (x \right )^{2}+2 x^{2}}+y \left (x \right )+x \right )}{y \left (x \right )-x}\right )+\sqrt {2}\, x -\sqrt {2}\, \ln \left (x +1\right )-\ln \left (x \right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 5.179 (sec). Leaf size: 81
DSolve[y'[x] == (y[x] + x*y[x] - x^2*Sqrt[x^2 + y[x]^2] + x*y[x]*Sqrt[x^2 + y[x]^2])/(x*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x \tanh \left (\frac {x-\log (x+1)+c_1}{\sqrt {2}}\right ) \left (2+\sqrt {2} \tanh \left (\frac {x-\log (x+1)+c_1}{\sqrt {2}}\right )\right )}{\sqrt {2}+2 \tanh \left (\frac {x-\log (x+1)+c_1}{\sqrt {2}}\right )} y(x)\to x \end{align*}