Internal problem ID [8256]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 676.
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 +1+2 x^{6} \sqrt {4 x^{2} y+1}}{2 x^{3} \left (x +1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.497 (sec). Leaf size: 43
dsolve(diff(y(x),x) = 1/2*(x+1+2*x^6*(4*x^2*y(x)+1)^(1/2))/x^3/(x+1),y(x), singsol=all)
\[ c_{1}+2 \ln \left (x +1\right )-\frac {\sqrt {4 x^{2} y \relax (x )+1}}{x}+\frac {x^{4}}{2}-\frac {2 x^{3}}{3}+x^{2}-2 x = 0 \]
✓ Solution by Mathematica
Time used: 5.706 (sec). Leaf size: 49
DSolve[y'[x] == (1/2 + x/2 + x^6*Sqrt[1 + 4*x^2*y[x]])/(x^3*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{4} \left (-\frac {1}{x^2}+\log ^2\left ((x+1)^2 \exp \left (\frac {1}{6} x (x (x (3 x-4)+6)-12)-2 c_1\right )\right )\right ) \\ \end{align*}