Internal problem ID [8467]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 887.
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 {y a^{2} x +a +a^{2} x +y^{3} a^{3} x^{3}+3 y^{2} a^{2} x^{2}+3 a y x +1}{a^{2} x^{2} \left (a y x +1+a x \right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 70
dsolve(diff(y(x),x) = 1/a^2/x^2*(y(x)*a^2*x+a+a^2*x+y(x)^3*a^3*x^3+3*y(x)^2*a^2*x^2+3*y(x)*a*x+1)/(y(x)*a*x+1+a*x),y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {-x a +\sqrt {c_{1}-2 x}-1}{x a \left (\sqrt {c_{1}-2 x}-1\right )} \\ y \relax (x ) = -\frac {x a +\sqrt {c_{1}-2 x}+1}{x a \left (\sqrt {c_{1}-2 x}+1\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.82 (sec). Leaf size: 99
DSolve[y'[x] == (1 + a + a^2*x + 3*a*x*y[x] + a^2*x*y[x] + 3*a^2*x^2*y[x]^2 + a^3*x^3*y[x]^3)/(a^2*x^2*(1 + a*x + a*x*y[x])),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{a x}+\frac {1}{-1+\frac {\sqrt {-2 a^6 x+c_1}}{a^3}} \\ y(x)\to -\frac {\sqrt {-2 a^6 x+c_1}+a^4 x+a^3}{a^4 x+a x \sqrt {-2 a^6 x+c_1}} \\ y(x)\to -\frac {1}{a x} \\ \end{align*}