60.2.194 problem 770

Internal problem ID [10781]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 770
Date solved : Monday, January 27, 2025 at 09:44:05 PM
CAS classification : [_rational]

\begin{align*} y^{\prime }&=\frac {2 y^{6}}{y^{3}+2+16 x y^{2}+32 x^{2} y^{4}} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 775

dsolve(diff(y(x),x) = 2*y(x)^6/(y(x)^3+2+16*x*y(x)^2+32*x^2*y(x)^4),y(x), singsol=all)
 

\begin{align*} y &= \frac {\left (4096 c_{1}^{3} x^{3}+6 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1} +256 x}\, c_{1} +96 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1} +256 x}\, x +54 c_{1}^{3}+1440 c_{1}^{2} x +9216 c_{1} x^{2}\right )^{{1}/{3}}+\frac {256 c_{1}^{2} x^{2}-12 c_{1} -192 x}{\left (4096 c_{1}^{3} x^{3}+6 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1} +256 x}\, c_{1} +96 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1} +256 x}\, x +54 c_{1}^{3}+1440 c_{1}^{2} x +9216 c_{1} x^{2}\right )^{{1}/{3}}}+16 c_{1} x}{3 c_{1} +48 x} \\ y &= \frac {\frac {\left (-i \sqrt {3}-1\right ) {\left (96 \sqrt {3}\, \left (\frac {c_{1}}{16}+x \right ) \sqrt {\left (4096 x^{3}+27\right ) c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1} +256 x}+\left (4096 x^{3}+54\right ) c_{1}^{3}+1440 c_{1}^{2} x +9216 c_{1} x^{2}\right )}^{{2}/{3}}}{6}+\frac {16 c_{1} x {\left (96 \sqrt {3}\, \left (\frac {c_{1}}{16}+x \right ) \sqrt {\left (4096 x^{3}+27\right ) c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1} +256 x}+\left (4096 x^{3}+54\right ) c_{1}^{3}+1440 c_{1}^{2} x +9216 c_{1} x^{2}\right )}^{{1}/{3}}}{3}+\frac {128 \left (c_{1}^{2} x^{2}-\frac {3}{4} x -\frac {3}{64} c_{1} \right ) \left (i \sqrt {3}-1\right )}{3}}{{\left (96 \sqrt {3}\, \left (\frac {c_{1}}{16}+x \right ) \sqrt {\left (4096 x^{3}+27\right ) c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1} +256 x}+\left (4096 x^{3}+54\right ) c_{1}^{3}+1440 c_{1}^{2} x +9216 c_{1} x^{2}\right )}^{{1}/{3}} \left (c_{1} +16 x \right )} \\ y &= \frac {\frac {\left (i \sqrt {3}-1\right ) {\left (96 \sqrt {3}\, \left (\frac {c_{1}}{16}+x \right ) \sqrt {\left (4096 x^{3}+27\right ) c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1} +256 x}+\left (4096 x^{3}+54\right ) c_{1}^{3}+1440 c_{1}^{2} x +9216 c_{1} x^{2}\right )}^{{2}/{3}}}{6}+\frac {16 c_{1} x {\left (96 \sqrt {3}\, \left (\frac {c_{1}}{16}+x \right ) \sqrt {\left (4096 x^{3}+27\right ) c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1} +256 x}+\left (4096 x^{3}+54\right ) c_{1}^{3}+1440 c_{1}^{2} x +9216 c_{1} x^{2}\right )}^{{1}/{3}}}{3}+\frac {128 \left (-i \sqrt {3}-1\right ) \left (c_{1}^{2} x^{2}-\frac {3}{4} x -\frac {3}{64} c_{1} \right )}{3}}{{\left (96 \sqrt {3}\, \left (\frac {c_{1}}{16}+x \right ) \sqrt {\left (4096 x^{3}+27\right ) c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1} +256 x}+\left (4096 x^{3}+54\right ) c_{1}^{3}+1440 c_{1}^{2} x +9216 c_{1} x^{2}\right )}^{{1}/{3}} \left (c_{1} +16 x \right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 19.606 (sec). Leaf size: 952

DSolve[D[y[x],x] == (2*y[x]^6)/(2 + 16*x*y[x]^2 + y[x]^3 + 32*x^2*y[x]^4),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {2 \sqrt [3]{2} \sqrt [3]{2048 x^3+4608 c_1{}^2 x^2+3 \sqrt {3} \sqrt {(1-16 c_1 x){}^2 \left (4096 x^3+2048 c_1{}^2 x^2+64 c_1 \left (-9+4 c_1{}^3\right ) x+27-16 c_1{}^3\right )}-720 c_1 x+27}+\frac {4\ 2^{2/3} \left (64 x^2-48 c_1{}^2 x+3 c_1\right )}{\sqrt [3]{2048 x^3+4608 c_1{}^2 x^2+3 \sqrt {3} \sqrt {(1-16 c_1 x){}^2 \left (4096 x^3+2048 c_1{}^2 x^2+64 c_1 \left (-9+4 c_1{}^3\right ) x+27-16 c_1{}^3\right )}-720 c_1 x+27}}+32 x}{6 (1-16 c_1 x)} \\ y(x)\to \frac {2 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \sqrt [3]{2048 x^3+4608 c_1{}^2 x^2+3 \sqrt {3} \sqrt {(1-16 c_1 x){}^2 \left (4096 x^3+2048 c_1{}^2 x^2+64 c_1 \left (-9+4 c_1{}^3\right ) x+27-16 c_1{}^3\right )}-720 c_1 x+27}-\frac {4 i 2^{2/3} \left (\sqrt {3}-i\right ) \left (64 x^2-48 c_1{}^2 x+3 c_1\right )}{\sqrt [3]{2048 x^3+4608 c_1{}^2 x^2+3 \sqrt {3} \sqrt {(1-16 c_1 x){}^2 \left (4096 x^3+2048 c_1{}^2 x^2+64 c_1 \left (-9+4 c_1{}^3\right ) x+27-16 c_1{}^3\right )}-720 c_1 x+27}}+64 x}{12 (1-16 c_1 x)} \\ y(x)\to \frac {-2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) \sqrt [3]{2048 x^3+4608 c_1{}^2 x^2+3 \sqrt {3} \sqrt {(1-16 c_1 x){}^2 \left (4096 x^3+2048 c_1{}^2 x^2+64 c_1 \left (-9+4 c_1{}^3\right ) x+27-16 c_1{}^3\right )}-720 c_1 x+27}+\frac {4 i 2^{2/3} \left (\sqrt {3}+i\right ) \left (64 x^2-48 c_1{}^2 x+3 c_1\right )}{\sqrt [3]{2048 x^3+4608 c_1{}^2 x^2+3 \sqrt {3} \sqrt {(1-16 c_1 x){}^2 \left (4096 x^3+2048 c_1{}^2 x^2+64 c_1 \left (-9+4 c_1{}^3\right ) x+27-16 c_1{}^3\right )}-720 c_1 x+27}}+64 x}{12 (1-16 c_1 x)} \\ y(x)\to 0 \\ y(x)\to \frac {x-\sqrt [3]{x^3}}{2 \sqrt {3} x \sqrt [6]{x^3}} \\ y(x)\to \frac {\sqrt [3]{x^3}-x}{2 \sqrt {3} x \sqrt [6]{x^3}} \\ y(x)\to \frac {\left (\sqrt {3}-3 i\right ) x-\left (\sqrt {3}+3 i\right ) \sqrt [3]{x^3}}{12 x \sqrt [6]{x^3}} \\ y(x)\to \frac {\left (\sqrt {3}+3 i\right ) x-\left (\sqrt {3}-3 i\right ) \sqrt [3]{x^3}}{12 x \sqrt [6]{x^3}} \\ y(x)\to \frac {\left (\sqrt {3}-3 i\right ) \sqrt [3]{x^3}-\left (\sqrt {3}+3 i\right ) x}{12 x \sqrt [6]{x^3}} \\ y(x)\to \frac {\left (\sqrt {3}+3 i\right ) \sqrt [3]{x^3}-\left (\sqrt {3}-3 i\right ) x}{12 x \sqrt [6]{x^3}} \\ \end{align*}