Internal problem ID [8289]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 709.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [y=_G(x,y')]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {2 a x +2 a +x^{3} \sqrt {-y^{2}+4 a x}}{\left (x +1\right ) y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 39
dsolve(diff(y(x),x) = (2*a*x+2*a+x^3*(-y(x)^2+4*a*x)^(1/2))/(x+1)/y(x),y(x), singsol=all)
\[ -\sqrt {-y \relax (x )^{2}+4 x a}-\frac {x^{3}}{3}+\frac {x^{2}}{2}-x +\ln \left (x +1\right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 3.705 (sec). Leaf size: 131
DSolve[y'[x] == (2*a + 2*a*x + x^3*Sqrt[4*a*x - y[x]^2])/((1 + x)*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{6} \sqrt {144 a x-(x (x (2 x-3)+6)+6 c_1){}^2+12 \log (x+1) (x (x (2 x-3)+6)-3 \log (x+1)+6 c_1)} \\ y(x)\to \frac {1}{6} \sqrt {144 a x-(x (x (2 x-3)+6)+6 c_1){}^2+12 \log (x+1) (x (x (2 x-3)+6)-3 \log (x+1)+6 c_1)} \\ \end{align*}