Internal problem ID [9000]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 665.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]
\[ \boxed {y^{\prime }+\frac {\left (\sqrt {a}\, x^{4}+x^{3} \sqrt {a}-2 \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2+2 x}=0} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 30
dsolve(diff(y(x),x) = -1/2*(a^(1/2)*x^4+a^(1/2)*x^3-2*(a*x^4+8*y(x))^(1/2))*a^(1/2)/(x+1),y(x), singsol=all)
\[ 4 \sqrt {a}\, \ln \left (x +1\right )-\sqrt {a \,x^{4}+8 y \left (x \right )}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.686 (sec). Leaf size: 39
DSolve[y'[x] == -1/2*(Sqrt[a]*(Sqrt[a]*x^3 + Sqrt[a]*x^4 - 2*Sqrt[a*x^4 + 8*y[x]]))/(1 + x),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {a x^4}{8}+2 a \log ^2(x+1)-4 a c_1 \log (x+1)+2 a c_1{}^2 \]