60.2.408 problem 985

Internal problem ID [10995]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 985
Date solved : Monday, January 27, 2025 at 10:39:43 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Abel]

\begin{align*} y^{\prime }&=\frac {\left (y x +1\right ) \left (x^{2} y^{2}+x^{2} y+2 y x +1+x +x^{2}\right )}{x^{5}} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 43

dsolve(diff(y(x),x) = (x*y(x)+1)*(x^2*y(x)^2+x^2*y(x)+2*x*y(x)+1+x+x^2)/x^5,y(x), singsol=all)
 

\[ y = \frac {17 \operatorname {RootOf}\left (162 \left (\int _{}^{\textit {\_Z}}\frac {1}{289 \textit {\_a}^{3}+54 \textit {\_a} -54}d \textit {\_a} \right ) x +3 c_{1} x +2\right ) x -3 x -9}{9 x} \]

✓ Solution by Mathematica

Time used: 0.247 (sec). Leaf size: 86

DSolve[D[y[x],x] == ((1 + x*y[x])*(1 + x + x^2 + 2*x*y[x] + x^2*y[x] + x^2*y[x]^2))/x^5,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{\frac {\frac {x+3}{x^3}+\frac {3 y(x)}{x^2}}{\sqrt [3]{34} \sqrt [3]{-\frac {1}{x^6}}}}\frac {1}{K[1]^3-\frac {3 \sqrt [3]{-2} K[1]}{17^{2/3}}+1}dK[1]=-\frac {1}{9} 34^{2/3} \left (-\frac {1}{x^6}\right )^{2/3} x^3+c_1,y(x)\right ] \]