Internal problem ID [8226]
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)]`]]
\[ \boxed {y^{\prime }+\frac {x^{3} \left (x \sqrt {a}+\sqrt {a}-2 \sqrt {x^{4} a +8 y}\right ) \sqrt {a}}{2 x +2}=0} \]
✓ Solution by Maple
Time used: 0.093 (sec). Leaf size: 44
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)
\[ -\sqrt {a \,x^{4}+8 y \left (x \right )}+4 \sqrt {a}\, \left (\frac {x^{3}}{3}-\frac {x^{2}}{2}+x -\ln \left (x +1\right )\right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.847 (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*}