Internal problem ID [8983]
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)]`]]
\[ \boxed {y^{\prime }+\frac {x^{3} \left (\sqrt {a}\, x +\sqrt {a}-2 \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2 x +2}=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 49
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)
\[ -\sqrt {x^{4} a +8 y \left (x \right )}-4 \sqrt {a}\, \ln \left (x +1\right )+\frac {2 \left (2 x^{3}-3 x^{2}+6 x \right ) \sqrt {a}}{3}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.823 (sec). Leaf size: 96
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]
\[ y(x)\to \frac {1}{72} a \left (16 x^6-48 x^5+123 x^4-96 c_1 x^3+72 (-1+2 c_1) x^2-48 \left (2 x^3-3 x^2+6 x+9-6 c_1\right ) \log (x+1)+144 \log ^2(x+1)-144 (-3+2 c_1) x+36 (3-2 c_1){}^2\right ) \]