60.2.300 problem 877

Internal problem ID [10887]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 877
Date solved : Monday, January 27, 2025 at 10:21:13 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class C`]]

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

✓ Solution by Maple

Time used: 0.037 (sec). Leaf size: 73

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

\begin{align*} y &= \frac {-2 c_{1} x^{2}+2 x^{3}+\sqrt {-2 x +2 c_{1} +1}-1}{-2 c_{1} +2 x} \\ y &= \frac {2 c_{1} x^{2}-2 x^{3}+\sqrt {-2 x +2 c_{1} +1}+1}{-2 x +2 c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.360 (sec). Leaf size: 54

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

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