Internal problem ID [8228]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 648.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)]]]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {x^{3} \left (\sqrt {a}\, x +\sqrt {a}-2 \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2 x +2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 50
dsolve(diff(y(x),x) = -1/2*x^3*(a^(1/2)*x+a^(1/2)-2*(a*x^4+8*y(x))^(1/2))*a^(1/2)/(x+1),y(x), singsol=all)
\[ \sqrt {a \,x^{4}+8 y \relax (x )}-\frac {4 x^{3} \sqrt {a}}{3}+2 \sqrt {a}\, x^{2}-4 \sqrt {a}\, x +4 \sqrt {a}\, \ln \left (x +1\right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.792 (sec). Leaf size: 56
DSolve[y'[x] == -1/2*(Sqrt[a]*x^3*(Sqrt[a] + Sqrt[a]*x - 2*Sqrt[a*x^4 + 8*y[x]]))/(1 + x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{72} a (x (x (4 x-9)+12)-12 \log (x+1)+18-12 c_1) (x (x (4 x-3)+12)-12 \log (x+1)+18-12 c_1) \\ \end{align*}