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 {y x +y+x^{4} \sqrt {y^{2}+x^{2}}}{x \left (x +1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.328 (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)
\[ \ln \left (\sqrt {x^{2}+y \relax (x )^{2}}+y \relax (x )\right )-\frac {x^{3}}{3}+\frac {x^{2}}{2}-x -\ln \relax (x )+\ln \left (x +1\right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 2.201 (sec). Leaf size: 72
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 e^{-\frac {1}{6} x (x (2 x+3)+6)-\frac {11}{6}-c_1} \left (-e^{x^2} (x+1)^2+e^{\frac {2 x^3}{3}+2 x+\frac {11}{3}+2 c_1}\right )}{2 (x+1)} \\ \end{align*}