Internal problem ID [8409]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 829.
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 {1+2 \sqrt {4 y x^{2}+1}\, x^{3}+2 x^{5} \sqrt {4 y x^{2}+1}+2 x^{6} \sqrt {4 y x^{2}+1}}{2 x^{3}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.531 (sec). Leaf size: 34
dsolve(diff(y(x),x) = 1/2*(1+2*(4*x^2*y(x)+1)^(1/2)*x^3+2*x^5*(4*x^2*y(x)+1)^(1/2)+2*x^6*(4*x^2*y(x)+1)^(1/2))/x^3,y(x), singsol=all)
\[ c_{1}-\frac {\sqrt {4 x^{2} y \relax (x )+1}}{x}+x^{2}+\frac {x^{4}}{2}+\frac {2 x^{5}}{5} = 0 \]
✓ Solution by Mathematica
Time used: 0.591 (sec). Leaf size: 56
DSolve[y'[x] == (1/2 + x^3*Sqrt[1 + 4*x^2*y[x]] + x^5*Sqrt[1 + 4*x^2*y[x]] + x^6*Sqrt[1 + 4*x^2*y[x]])/x^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\left (4 x^6+5 x^5+10 x^3-20 c_1 x-10\right ) \left (4 x^6+5 x^5+10 x^3-20 c_1 x+10\right )}{400 x^2} \\ \end{align*}