60.2.354 problem 931

Internal problem ID [10941]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 931
Date solved : Monday, January 27, 2025 at 10:26:06 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class C`], [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=\frac {-3 x^{2} y-2 x^{3}-2 x -x y^{2}-y+x^{3} y^{3}+3 x^{4} y^{2}+3 x^{5} y+x^{6}}{x \left (y x +x^{2}+1\right )} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 73

dsolve(diff(y(x),x) = (-3*x^2*y(x)-2*x^3-2*x-x*y(x)^2-y(x)+x^3*y(x)^3+3*x^4*y(x)^2+3*x^5*y(x)+x^6)/x/(x*y(x)+x^2+1),y(x), singsol=all)
 

\begin{align*} y &= \frac {-\sqrt {c_{1} -2 x}\, x^{2}+x^{2}+1}{\left (\sqrt {c_{1} -2 x}-1\right ) x} \\ y &= \frac {-\sqrt {c_{1} -2 x}\, x^{2}-x^{2}-1}{\left (\sqrt {c_{1} -2 x}+1\right ) x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.404 (sec). Leaf size: 60

DSolve[D[y[x],x] == (-2*x - 2*x^3 + x^6 - y[x] - 3*x^2*y[x] + 3*x^5*y[x] - x*y[x]^2 + 3*x^4*y[x]^2 + x^3*y[x]^3)/(x*(1 + x^2 + x*y[x])),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -x+\frac {1}{x \left (-1+\sqrt {-2 x+c_1}\right )} \\ y(x)\to -x-\frac {1}{x+x \sqrt {-2 x+c_1}} \\ y(x)\to -x \\ \end{align*}