Internal problem ID [8488]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 908.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {4 x \left (a -1\right ) \left (a +1\right )}{4 y+y^{4} a^{2}-2 a^{4} y^{2} x^{2}+4 y^{2} a^{2} x^{2}+a^{6} x^{4}-3 a^{4} x^{4}+3 a^{2} x^{4}-y^{4}-2 y^{2} x^{2}-x^{4}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 1742
dsolve(diff(y(x),x) = 4*x*(a-1)*(a+1)/(4*y(x)+a^2*y(x)^4-2*a^4*y(x)^2*x^2+4*y(x)^2*a^2*x^2+a^6*x^4-3*a^4*x^4+3*a^2*x^4-y(x)^4-2*x^2*y(x)^2-x^4),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (\left (-c_{1} a^{2}+c_{1}\right ) 9^{\frac {1}{3}} \left (\left (\frac {\sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}}{3}+3+\left (-\frac {a^{2}}{9}+\frac {1}{9}\right ) c_{1}^{3}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1}\right ) \left (a +1\right )^{2} \left (a -1\right )^{2}\right )^{\frac {1}{3}}+3 a^{6} x^{2}+\left (-9 x^{2}+c_{1}^{2}\right ) a^{4}+\left (9 x^{2}-2 c_{1}^{2}\right ) a^{2}+\left (\left (9 c_{1} a^{4} x^{2}-c_{1}^{3} a^{2}-18 c_{1} a^{2} x^{2}+c_{1}^{3}+9 x^{2} c_{1}+3 \sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}+27\right ) \left (a^{2}-1\right )^{2}\right )^{\frac {2}{3}}-3 x^{2}+c_{1}^{2}\right ) 9^{\frac {2}{3}}}{9 \left (\left (\frac {\sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}}{3}+3+\left (-\frac {a^{2}}{9}+\frac {1}{9}\right ) c_{1}^{3}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1}\right ) \left (a +1\right )^{2} \left (a -1\right )^{2}\right )^{\frac {1}{3}} \left (3 a^{2}-3\right )} \\ y \relax (x ) = -\frac {\left (\left (2 c_{1} a^{2}-2 c_{1}\right ) 9^{\frac {1}{3}} \left (\left (\frac {\sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}}{3}+3+\left (-\frac {a^{2}}{9}+\frac {1}{9}\right ) c_{1}^{3}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1}\right ) \left (a +1\right )^{2} \left (a -1\right )^{2}\right )^{\frac {1}{3}}+\left (-3 i a^{6} x^{2}+\left (9 i x^{2}-i c_{1}^{2}\right ) a^{4}+\left (-9 i x^{2}+2 i c_{1}^{2}\right ) a^{2}+i \left (\left (9 c_{1} a^{4} x^{2}-c_{1}^{3} a^{2}-18 c_{1} a^{2} x^{2}+c_{1}^{3}+9 x^{2} c_{1}+3 \sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}+27\right ) \left (a^{2}-1\right )^{2}\right )^{\frac {2}{3}}+3 i x^{2}-i c_{1}^{2}\right ) \sqrt {3}+3 a^{6} x^{2}+\left (-9 x^{2}+c_{1}^{2}\right ) a^{4}+\left (9 x^{2}-2 c_{1}^{2}\right ) a^{2}+\left (\left (9 c_{1} a^{4} x^{2}-c_{1}^{3} a^{2}-18 c_{1} a^{2} x^{2}+c_{1}^{3}+9 x^{2} c_{1}+3 \sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}+27\right ) \left (a^{2}-1\right )^{2}\right )^{\frac {2}{3}}-3 x^{2}+c_{1}^{2}\right ) 9^{\frac {2}{3}}}{9 \left (\left (\frac {\sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}}{3}+3+\left (-\frac {a^{2}}{9}+\frac {1}{9}\right ) c_{1}^{3}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1}\right ) \left (a +1\right )^{2} \left (a -1\right )^{2}\right )^{\frac {1}{3}} \left (6 a^{2}-6\right )} \\ y \relax (x ) = \frac {\left (\left (-2 c_{1} a^{2}+2 c_{1}\right ) 9^{\frac {1}{3}} \left (\left (\frac {\sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}}{3}+3+\left (-\frac {a^{2}}{9}+\frac {1}{9}\right ) c_{1}^{3}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1}\right ) \left (a +1\right )^{2} \left (a -1\right )^{2}\right )^{\frac {1}{3}}+\left (-3 i a^{6} x^{2}+\left (9 i x^{2}-i c_{1}^{2}\right ) a^{4}+\left (-9 i x^{2}+2 i c_{1}^{2}\right ) a^{2}+i \left (\left (9 c_{1} a^{4} x^{2}-c_{1}^{3} a^{2}-18 c_{1} a^{2} x^{2}+c_{1}^{3}+9 x^{2} c_{1}+3 \sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}+27\right ) \left (a^{2}-1\right )^{2}\right )^{\frac {2}{3}}+3 i x^{2}-i c_{1}^{2}\right ) \sqrt {3}-3 a^{6} x^{2}+\left (9 x^{2}-c_{1}^{2}\right ) a^{4}+\left (-9 x^{2}+2 c_{1}^{2}\right ) a^{2}-\left (\left (9 c_{1} a^{4} x^{2}-c_{1}^{3} a^{2}-18 c_{1} a^{2} x^{2}+c_{1}^{3}+9 x^{2} c_{1}+3 \sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}+27\right ) \left (a^{2}-1\right )^{2}\right )^{\frac {2}{3}}+3 x^{2}-c_{1}^{2}\right ) 9^{\frac {2}{3}}}{9 \left (\left (\frac {\sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}}{3}+3+\left (-\frac {a^{2}}{9}+\frac {1}{9}\right ) c_{1}^{3}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1}\right ) \left (a +1\right )^{2} \left (a -1\right )^{2}\right )^{\frac {1}{3}} \left (6 a^{2}-6\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 16.724 (sec). Leaf size: 915
DSolve[y'[x] == (4*(-1 + a)*(1 + a)*x)/(-x^4 + 3*a^2*x^4 - 3*a^4*x^4 + a^6*x^4 + 4*y[x] - 2*x^2*y[x]^2 + 4*a^2*x^2*y[x]^2 - 2*a^4*x^2*y[x]^2 - y[x]^4 + a^2*y[x]^4),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{-9 a^6 c_1 x^2+27 a^4 \left (1+c_1 x^2\right )-27 a^2 \left (2+c_1 x^2\right )+\frac {1}{2} \sqrt {4 \left (-9 \left (a^2-1\right )^3 c_1 x^2+27 \left (a^2-1\right )^2+c_1{}^3\right ){}^2-4 \left (3 \left (a^2-1\right )^3 x^2+c_1{}^2\right ){}^3}+9 c_1 x^2+27+c_1{}^3}+\frac {3 \left (a^2-1\right )^3 x^2+c_1{}^2}{\sqrt [3]{-9 a^6 c_1 x^2+27 a^4 \left (1+c_1 x^2\right )-27 a^2 \left (2+c_1 x^2\right )+\frac {1}{2} \sqrt {4 \left (-9 \left (a^2-1\right )^3 c_1 x^2+27 \left (a^2-1\right )^2+c_1{}^3\right ){}^2-4 \left (3 \left (a^2-1\right )^3 x^2+c_1{}^2\right ){}^3}+9 c_1 x^2+27+c_1{}^3}}+c_1}{3 \left (a^2-1\right )} \\ y(x)\to \frac {2 i \left (\sqrt {3}+i\right ) \sqrt [3]{-9 a^6 c_1 x^2+27 a^4 \left (1+c_1 x^2\right )-27 a^2 \left (2+c_1 x^2\right )+\frac {1}{2} \sqrt {4 \left (-9 \left (a^2-1\right )^3 c_1 x^2+27 \left (a^2-1\right )^2+c_1{}^3\right ){}^2-4 \left (3 \left (a^2-1\right )^3 x^2+c_1{}^2\right ){}^3}+9 c_1 x^2+27+c_1{}^3}-\frac {2 i \left (\sqrt {3}-i\right ) \left (3 \left (a^2-1\right )^3 x^2+c_1{}^2\right )}{\sqrt [3]{-9 a^6 c_1 x^2+27 a^4 \left (1+c_1 x^2\right )-27 a^2 \left (2+c_1 x^2\right )+\frac {1}{2} \sqrt {4 \left (-9 \left (a^2-1\right )^3 c_1 x^2+27 \left (a^2-1\right )^2+c_1{}^3\right ){}^2-4 \left (3 \left (a^2-1\right )^3 x^2+c_1{}^2\right ){}^3}+9 c_1 x^2+27+c_1{}^3}}+4 c_1}{12 \left (a^2-1\right )} \\ y(x)\to \frac {-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{-9 a^6 c_1 x^2+27 a^4 \left (1+c_1 x^2\right )-27 a^2 \left (2+c_1 x^2\right )+\frac {1}{2} \sqrt {4 \left (-9 \left (a^2-1\right )^3 c_1 x^2+27 \left (a^2-1\right )^2+c_1{}^3\right ){}^2-4 \left (3 \left (a^2-1\right )^3 x^2+c_1{}^2\right ){}^3}+9 c_1 x^2+27+c_1{}^3}+\frac {2 i \left (\sqrt {3}+i\right ) \left (3 \left (a^2-1\right )^3 x^2+c_1{}^2\right )}{\sqrt [3]{-9 a^6 c_1 x^2+27 a^4 \left (1+c_1 x^2\right )-27 a^2 \left (2+c_1 x^2\right )+\frac {1}{2} \sqrt {4 \left (-9 \left (a^2-1\right )^3 c_1 x^2+27 \left (a^2-1\right )^2+c_1{}^3\right ){}^2-4 \left (3 \left (a^2-1\right )^3 x^2+c_1{}^2\right ){}^3}+9 c_1 x^2+27+c_1{}^3}}+4 c_1}{12 \left (a^2-1\right )} \\ y(x)\to -\frac {i \sqrt {-\left (a^2-1\right )^3 x^2}}{a^2-1} \\ y(x)\to \frac {i \sqrt {-\left (a^2-1\right )^3 x^2}}{a^2-1} \\ \end{align*}