29.20.17 problem 564

Internal problem ID [5158]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 20
Problem number : 564
Date solved : Monday, January 27, 2025 at 10:16:28 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class C`]]

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

✓ Solution by Maple

Time used: 0.247 (sec). Leaf size: 107

dsolve(x*(x^n+a*y(x))*diff(y(x),x)+(b+c*y(x))*y(x)^2 = 0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.960 (sec). Leaf size: 91

DSolve[x(x^n+a y[x])D[y[x],x]+(b+c y[x])y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\frac {y(x)^{-\frac {a n+b}{b}} (c y(x)-a n) (b+c y(x))^{\frac {a n}{b}}}{a^2 n^2 (a n+b)}-\frac {x^{-n} e^{-\frac {a n (\log (y(x))-\log (b+c y(x)))}{b}}}{a n^2}=c_1,y(x)\right ] \]