Internal problem ID [8459]
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)]]]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {-x y-y+x^{2} \sqrt {x^{2}+y^{2}}-x \sqrt {x^{2}+y^{2}}\, y}{x \left (x +1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.129 (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)
\[ \ln \left (\frac {2 x \left (\sqrt {2 x^{2}+2 y \relax (x )^{2}}+y \relax (x )+x \right )}{-x +y \relax (x )}\right )+x \sqrt {2}-\sqrt {2}\, \ln \left (x +1\right )-\ln \relax (x )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 1.884 (sec). Leaf size: 69
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 {1}{2} x \left (\sqrt {2} \tanh \left (\frac {x-\log (x+1)+c_1}{\sqrt {2}}\right )-\frac {1}{1+\sqrt {2} \tanh \left (\frac {x-\log (x+1)+c_1}{\sqrt {2}}\right )}+1\right ) \\ y(x)\to x \\ \end{align*}