60.2.349 problem 926

Internal problem ID [10936]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 926
Date solved : Monday, January 27, 2025 at 10:25:41 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {-8 x^{2} y^{3}+16 x y^{2}+16 x y^{3}-8+12 y x -6 x^{2} y^{2}+x^{3} y^{3}}{16 \left (-2+y x -2 y\right ) x} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 67

dsolve(diff(y(x),x) = 1/16*(-8*x^2*y(x)^3+16*x*y(x)^2+16*x*y(x)^3-8+12*x*y(x)-6*x^2*y(x)^2+x^3*y(x)^3)/(-2+x*y(x)-2*y(x))/x,y(x), singsol=all)
 

\begin{align*} y &= \frac {2 \sqrt {c_{1} +8 \ln \left (x \right )}+8}{x \sqrt {c_{1} +8 \ln \left (x \right )}+4 x -8} \\ y &= \frac {2 \sqrt {c_{1} +8 \ln \left (x \right )}-8}{x \sqrt {c_{1} +8 \ln \left (x \right )}-4 x +8} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.765 (sec). Leaf size: 650

DSolve[D[y[x],x] == (-1/2 + (3*x*y[x])/4 + x*y[x]^2 - (3*x^2*y[x]^2)/8 + x*y[x]^3 - (x^2*y[x]^3)/2 + (x^3*y[x]^3)/16)/(x*(-2 - 2*y[x] + x*y[x])),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {32-\frac {64 \sqrt {-2 \int _1^x-\frac {1024 \exp \left (2 \int _1^{K[2]}\frac {2}{(K[1]-2) K[1]}dK[1]\right ) K[2]}{(K[2]-2)^2}dK[2]+c_1}}{x \left (64 \exp \left (\int _1^x\frac {2}{(K[1]-2) K[1]}dK[1]\right )+\sqrt {-2 \int _1^x-\frac {1024 \exp \left (2 \int _1^{K[2]}\frac {2}{(K[1]-2) K[1]}dK[1]\right ) K[2]}{(K[2]-2)^2}dK[2]+c_1}\right )}}{16 (x-2)} \\ y(x)\to \frac {32-\frac {64 \sqrt {-2 \int _1^x-\frac {1024 \exp \left (2 \int _1^{K[2]}\frac {2}{(K[1]-2) K[1]}dK[1]\right ) K[2]}{(K[2]-2)^2}dK[2]+c_1}}{x \left (-64 \exp \left (\int _1^x\frac {2}{(K[1]-2) K[1]}dK[1]\right )+\sqrt {-2 \int _1^x-\frac {1024 \exp \left (2 \int _1^{K[2]}\frac {2}{(K[1]-2) K[1]}dK[1]\right ) K[2]}{(K[2]-2)^2}dK[2]+c_1}\right )}}{16 (x-2)} \\ y(x)\to \frac {2}{x} \\ y(x)\to \frac {2 \left (\sqrt {2} (x-2) \sqrt {-\int _1^x-\frac {1024 \exp \left (2 \int _1^{K[2]}\frac {2}{(K[1]-2) K[1]}dK[1]\right ) K[2]}{(K[2]-2)^2}dK[2]}-64 x \exp \left (\int _1^x\frac {2}{(K[1]-2) K[1]}dK[1]\right )\right )}{(x-2) x \left (\sqrt {2} \sqrt {-\int _1^x-\frac {1024 \exp \left (2 \int _1^{K[2]}\frac {2}{(K[1]-2) K[1]}dK[1]\right ) K[2]}{(K[2]-2)^2}dK[2]}-64 \exp \left (\int _1^x\frac {2}{(K[1]-2) K[1]}dK[1]\right )\right )} \\ y(x)\to \frac {2 \left (\sqrt {2} (x-2) \sqrt {-\int _1^x-\frac {1024 \exp \left (2 \int _1^{K[2]}\frac {2}{(K[1]-2) K[1]}dK[1]\right ) K[2]}{(K[2]-2)^2}dK[2]}+64 x \exp \left (\int _1^x\frac {2}{(K[1]-2) K[1]}dK[1]\right )\right )}{(x-2) x \left (\sqrt {2} \sqrt {-\int _1^x-\frac {1024 \exp \left (2 \int _1^{K[2]}\frac {2}{(K[1]-2) K[1]}dK[1]\right ) K[2]}{(K[2]-2)^2}dK[2]}+64 \exp \left (\int _1^x\frac {2}{(K[1]-2) K[1]}dK[1]\right )\right )} \\ \end{align*}