Internal problem ID [7828]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 249.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class C`]]
\[ \boxed {\left (a x y+b \,x^{n}\right ) y^{\prime }+\alpha y^{3}+\beta y^{2}=0} \]
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 202
dsolve((a*x*y(x)+b*x^n)*diff(y(x),x)+alpha*y(x)^3+beta*y(x)^2=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\beta }{\operatorname {RootOf}\left (-x^{-n +1} \textit {\_Z}^{\frac {a \left (n -1\right )}{\beta }} a^{2} \beta n +c_{1} a^{2} b \,n^{2}+x^{-n +1} \textit {\_Z}^{\frac {a \left (n -1\right )}{\beta }} a^{2} \beta -x^{-n +1} \textit {\_Z}^{\frac {a \left (n -1\right )}{\beta }} a \,\beta ^{2}-\textit {\_Z}^{\frac {a n -a +\beta }{\beta }} \beta b a n +\textit {\_Z}^{\frac {a \left (n -1\right )}{\beta }} a \alpha b n -2 c_{1} a^{2} b n +c_{1} a b \beta n +\textit {\_Z}^{\frac {a n -a +\beta }{\beta }} \beta b a -\textit {\_Z}^{\frac {a \left (n -1\right )}{\beta }} a \alpha b +\textit {\_Z}^{\frac {a \left (n -1\right )}{\beta }} \alpha b \beta +a^{2} b c_{1} -c_{1} a b \beta \right ) \beta -\alpha } \]
✓ Solution by Mathematica
Time used: 2.546 (sec). Leaf size: 115
DSolve[(a*x*y[x]+b*x^n)*y'[x]+\[Alpha]*y[x]^3+\[Beta]*y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {(a (-n)+a+\alpha y(x)) y(x)^{\frac {a-a n}{\beta }-1} (\beta +\alpha y(x))^{\frac {a (n-1)}{\beta }}}{a^2 (n-1)^2 (a (n-1)+\beta )}+\frac {x^{1-n} \exp \left (-\frac {a (n-1) (\log (y(x))-\log (\beta +\alpha y(x)))}{\beta }\right )}{a b (1-n) (n-1)}=c_1,y(x)\right ] \]