60.2.331 problem 908
Internal
problem
ID
[10918]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
908
Date
solved
:
Tuesday, January 28, 2025 at 05:30:58 PM
CAS
classification
:
[_rational]
\begin{align*} y^{\prime }&=\frac {4 x \left (a -1\right ) \left (a +1\right )}{4 y+a^{2} y^{4}-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 x^{2} y^{2}-x^{4}} \end{align*}
✓ Solution by Maple
Time used: 0.011 (sec). Leaf size: 1345
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 &= \frac {\left (9^{{2}/{3}} \left (\left (a +1\right )^{2} \left (3+\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}+\frac {\left (-a^{2}+1\right ) c_{1}^{3}}{9}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1} \right ) \left (a -1\right )^{2}\right )^{{2}/{3}}+\left (-a^{2} c_{1} +c_{1} \right ) 9^{{1}/{3}} \left (\left (a +1\right )^{2} \left (3+\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}+\frac {\left (-a^{2}+1\right ) c_{1}^{3}}{9}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1} \right ) \left (a -1\right )^{2}\right )^{{1}/{3}}+3 a^{6} x^{2}+\left (c_{1}^{2}-9 x^{2}\right ) a^{4}+\left (-2 c_{1}^{2}+9 x^{2}\right ) a^{2}-3 x^{2}+c_{1}^{2}\right ) 9^{{2}/{3}}}{9 \left (\left (a +1\right )^{2} \left (3+\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}+\frac {\left (-a^{2}+1\right ) c_{1}^{3}}{9}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1} \right ) \left (a -1\right )^{2}\right )^{{1}/{3}} \left (3 a^{2}-3\right )} \\
y &= \frac {\left (\left (-\frac {i \sqrt {3}}{3}-\frac {1}{3}\right ) 9^{{2}/{3}} \left (\left (a +1\right )^{2} \left (3+\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}+\frac {\left (-a^{2}+1\right ) c_{1}^{3}}{9}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1} \right ) \left (a -1\right )^{2}\right )^{{2}/{3}}+\left (a +1\right ) \left (a -1\right ) \left (-\frac {2 \,9^{{1}/{3}} \left (\left (a +1\right )^{2} \left (3+\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}+\frac {\left (-a^{2}+1\right ) c_{1}^{3}}{9}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1} \right ) \left (a -1\right )^{2}\right )^{{1}/{3}} c_{1}}{3}+\left (a +1\right ) \left (a -1\right ) \left (a^{2} x^{2}-x^{2}+\frac {1}{3} c_{1}^{2}\right ) \left (i \sqrt {3}-1\right )\right )\right ) 9^{{2}/{3}}}{3 \left (\left (a +1\right )^{2} \left (3+\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}+\frac {\left (-a^{2}+1\right ) c_{1}^{3}}{9}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1} \right ) \left (a -1\right )^{2}\right )^{{1}/{3}} \left (6 a^{2}-6\right )} \\
y &= -\frac {\left (\left (-\frac {i \sqrt {3}}{3}+\frac {1}{3}\right ) 9^{{2}/{3}} \left (\left (a +1\right )^{2} \left (3+\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}+\frac {\left (-a^{2}+1\right ) c_{1}^{3}}{9}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1} \right ) \left (a -1\right )^{2}\right )^{{2}/{3}}+\left (\frac {2 \,9^{{1}/{3}} \left (\left (a +1\right )^{2} \left (3+\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}+\frac {\left (-a^{2}+1\right ) c_{1}^{3}}{9}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1} \right ) \left (a -1\right )^{2}\right )^{{1}/{3}} c_{1}}{3}+\left (a +1\right ) \left (1+i \sqrt {3}\right ) \left (a -1\right ) \left (a^{2} x^{2}-x^{2}+\frac {1}{3} c_{1}^{2}\right )\right ) \left (a +1\right ) \left (a -1\right )\right ) 9^{{2}/{3}}}{3 \left (\left (a +1\right )^{2} \left (3+\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}+\frac {\left (-a^{2}+1\right ) c_{1}^{3}}{9}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1} \right ) \left (a -1\right )^{2}\right )^{{1}/{3}} \left (6 a^{2}-6\right )} \\
\end{align*}
✓ Solution by Mathematica
Time used: 10.427 (sec). Leaf size: 1065
DSolve[D[y[x],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 c_1 x^2+27 a^4-27 a^2 c_1 x^2-54 a^2+\frac {1}{2} \sqrt {4 \left (-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 )+9 c_1 x^2+27+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 c_1 x^2+27 a^4-27 a^2 c_1 x^2-54 a^2+\frac {1}{2} \sqrt {4 \left (-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 )+9 c_1 x^2+27+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 c_1 x^2+27 a^4-27 a^2 c_1 x^2-54 a^2+\frac {1}{2} \sqrt {4 \left (-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 )+9 c_1 x^2+27+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 c_1 x^2+27 a^4-27 a^2 c_1 x^2-54 a^2+\frac {1}{2} \sqrt {4 \left (-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 )+9 c_1 x^2+27+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 c_1 x^2+27 a^4-27 a^2 c_1 x^2-54 a^2+\frac {1}{2} \sqrt {4 \left (-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 )+9 c_1 x^2+27+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 c_1 x^2+27 a^4-27 a^2 c_1 x^2-54 a^2+\frac {1}{2} \sqrt {4 \left (-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 )+9 c_1 x^2+27+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*}