Internal problem ID [8385]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 805.
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^{4} \sqrt {x^{2}+y^{2}}}{x \left (x +1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.27 (sec). Leaf size: 42
dsolve(diff(y(x),x) = (x*y(x)+y(x)+x^4*(y(x)^2+x^2)^(1/2))/x/(x+1),y(x), singsol=all)
\[ \ln \left (\sqrt {x^{2}+y \relax (x )^{2}}+y \relax (x )\right )-\frac {x^{3}}{3}+\frac {x^{2}}{2}-x +\ln \left (x +1\right )-\ln \relax (x )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 48.275 (sec). Leaf size: 138
DSolve[y'[x] == (y[x] + x*y[x] + x^4*Sqrt[x^2 + y[x]^2])/(x*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x \tanh \left (\frac {1}{6} (2 x-3) x^2+x-\log (x+1)+\frac {11}{6}+c_1\right )}{\sqrt {1-\tanh ^2\left (\frac {1}{6} (2 x-3) x^2+x-\log (x+1)+\frac {11}{6}+c_1\right )}} \\ y(x)\to \frac {x \tanh \left (\frac {1}{6} (2 x-3) x^2+x-\log (x+1)+\frac {11}{6}+c_1\right )}{\sqrt {1-\tanh ^2\left (\frac {1}{6} (2 x-3) x^2+x-\log (x+1)+\frac {11}{6}+c_1\right )}} \\ \end{align*}