Internal problem ID [8475]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 895.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_1st_order, _with_symmetry_[F(x),G(x)]], [_Abel, 2nd type, class C]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\left (-256 a \,x^{2} y-32 x^{6} a^{2}-256 x^{2} a +512 y^{3}+192 y^{2} a \,x^{4}+24 y a^{2} x^{8}+a^{3} x^{12}\right ) x}{512 y+64 x^{4} a +512}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 80
dsolve(diff(y(x),x) = (-256*a*x^2*y(x)-32*a^2*x^6-256*a*x^2+512*y(x)^3+192*x^4*a*y(x)^2+24*y(x)*a^2*x^8+a^3*x^12)*x/(512*y(x)+64*a*x^4+512),y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {\sqrt {-x^{2}+c_{1}}\, a \,x^{4}-a \,x^{4}-8}{8 \left (\sqrt {-x^{2}+c_{1}}-1\right )} \\ y \relax (x ) = -\frac {\sqrt {-x^{2}+c_{1}}\, a \,x^{4}+a \,x^{4}+8}{8 \left (\sqrt {-x^{2}+c_{1}}+1\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.458 (sec). Leaf size: 75
DSolve[y'[x] == (x*(-256*a*x^2 - 32*a^2*x^6 + a^3*x^12 - 256*a*x^2*y[x] + 24*a^2*x^8*y[x] + 192*a*x^4*y[x]^2 + 512*y[x]^3))/(512 + 64*a*x^4 + 512*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {a x^4}{8}+\frac {512}{-512+\sqrt {-262144 x^2+c_1}} \\ y(x)\to -\frac {a x^4}{8}-\frac {512}{512+\sqrt {-262144 x^2+c_1}} \\ y(x)\to -\frac {a x^4}{8} \\ \end{align*}