60.1.248 problem 249

Internal problem ID [10262]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 249
Date solved : Monday, January 27, 2025 at 06:42:20 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class C`]]

\begin{align*} \left (a x y+b \,x^{n}\right ) y^{\prime }+\alpha y^{3}+\beta y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.240 (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)
 

\[ y = \frac {\beta }{\operatorname {RootOf}\left (-x^{-n +1} \textit {\_Z}^{\frac {a \left (n -1\right )}{\beta }} a^{2} \beta n +c_{1} a^{2} b \,n^{2}+x^{-n +1} \textit {\_Z}^{\frac {a \left (n -1\right )}{\beta }} a^{2} \beta -x^{-n +1} \textit {\_Z}^{\frac {a \left (n -1\right )}{\beta }} a \,\beta ^{2}-\textit {\_Z}^{\frac {a n -a +\beta }{\beta }} \beta b a n +\textit {\_Z}^{\frac {a \left (n -1\right )}{\beta }} a \alpha b n -2 c_{1} a^{2} b n +c_{1} a b \beta n +\textit {\_Z}^{\frac {a n -a +\beta }{\beta }} \beta b a -\textit {\_Z}^{\frac {a \left (n -1\right )}{\beta }} a \alpha b +\textit {\_Z}^{\frac {a \left (n -1\right )}{\beta }} \alpha b \beta +c_{1} a^{2} b -c_{1} a b \beta \right ) \beta -\alpha } \]

✓ Solution by Mathematica

Time used: 2.300 (sec). Leaf size: 299

DSolve[(a*x*y[x]+b*x^n)*D[y[x],x]+\[Alpha]*y[x]^3+\[Beta]*y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{-\frac {b x^{n-1} \left (b (a (n-1)-\beta ) x^n+\left (-3 b \alpha x^n+a^2 (n-1) x+2 a \beta x\right ) y(x)\right )}{a \sqrt [3]{-\frac {b^3 x^{3 n-3} \left (2 a^3 (n-1)^3+3 a^2 \beta (n-1)^2-3 a \beta ^2 (n-1)-2 \beta ^3\right )}{a^3}} \left (b x^n+a y(x) x\right )}}\frac {1}{K[1]^3+\frac {3 \sqrt [3]{-1} \left (a^2 (n-1)^2+a \beta (n-1)+\beta ^2\right ) K[1]}{(a (n-1)-\beta )^{2/3} (2 a (n-1)+\beta )^{2/3} (a (n-1)+2 \beta )^{2/3}}+1}dK[1]+\frac {a x^{2-2 n} \left (-\frac {b^3 x^{3 n-3} \left (2 a^3 (n-1)^3+3 a^2 \beta (n-1)^2-3 a \beta ^2 (n-1)-2 \beta ^3\right )}{a^3}\right )^{2/3} \log \left (\alpha b-a \beta x^{1-n}\right )}{9 b^2 \beta (n-1)}=c_1,y(x)\right ] \]