60.2.311 problem 888

Internal problem ID [10898]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 888
Date solved : Monday, January 27, 2025 at 10:21:56 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 {6 x^{2} y-2 x +1-5 x^{3} y^{2}-2 y x +y^{3} x^{4}}{x^{2} \left (x^{2} y-x +1\right )} \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 79

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

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

✓ Solution by Mathematica

Time used: 0.785 (sec). Leaf size: 74

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

\begin{align*} y(x)\to \frac {x-1}{x^2}+\frac {1}{x^4 \left (\frac {1}{x^2}-\frac {1}{x^2 \sqrt {\frac {2}{x}+c_1}}\right )} \\ y(x)\to \frac {x+\frac {1}{1+\frac {1}{\sqrt {\frac {2}{x}+c_1}}}-1}{x^2} \\ y(x)\to \frac {1}{x} \\ \end{align*}