60.2.387 problem 964
Internal
problem
ID
[10974]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
964
Date
solved
:
Tuesday, January 28, 2025 at 05:40:06 PM
CAS
classification
:
[_rational]
\begin{align*} y^{\prime }&=-\frac {8 x \left (a -1\right ) \left (a +1\right )}{8+x^{6}+2 x^{4}+3 a^{4} y^{4} x^{2}-3 a^{6} y^{2} x^{4}+9 y^{2} a^{4} x^{4}-9 y^{2} a^{2} x^{4}-8 y-4 a^{2} x^{6}+4 a^{4} y^{2} x^{2}+y^{6}-8 y^{2} a^{2} x^{2}+3 x^{4} y^{2}+3 x^{2} y^{4}+2 y^{4}-8 a^{2}-6 y^{4} a^{2} x^{2}+4 x^{2} y^{2}-a^{2} y^{6}+a^{8} x^{6}-4 a^{6} x^{6}+6 a^{4} x^{6}-2 a^{2} y^{4}-2 a^{6} x^{4}+6 a^{4} x^{4}-6 a^{2} x^{4}} \end{align*}
✓ Solution by Maple
Time used: 1.651 (sec). Leaf size: 567
dsolve(diff(y(x),x) = -8*x*(a-1)*(a+1)/(8-a^2*y(x)^6+a^8*x^6-4*a^6*x^6+6*a^4*x^6-2*a^2*y(x)^4-2*a^6*x^4+6*a^4*x^4-6*a^2*x^4-4*a^2*x^6+y(x)^6+x^6-8*a^2+3*a^4*y(x)^4*x^2-3*a^6*y(x)^2*x^4+9*y(x)^2*a^4*x^4-9*y(x)^2*a^2*x^4+4*a^4*y(x)^2*x^2-6*y(x)^4*a^2*x^2-8*y(x)^2*a^2*x^2+3*x^2*y(x)^4+3*x^4*y(x)^2+4*x^2*y(x)^2+2*x^4+2*y(x)^4-8*y(x)),y(x), singsol=all)
\begin{align*}
y &= -\frac {2^{{5}/{6}} \sqrt {3}\, \sqrt {\left (3 a^{2} x^{2}-3 x^{2}-2\right ) \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}-\left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}-4}}{6 \left (3 \sqrt {3}+\sqrt {31}\right )^{{1}/{3}}} \\
y &= \frac {2^{{5}/{6}} \sqrt {3}\, \sqrt {\left (3 a^{2} x^{2}-3 x^{2}-2\right ) \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}-\left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}-4}}{6 \left (3 \sqrt {3}+\sqrt {31}\right )^{{1}/{3}}} \\
y &= -\frac {2^{{1}/{3}} \sqrt {3}\, \sqrt {\left (-4+\left (6 a^{2}-6\right ) x^{2}\right ) \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}-i \sqrt {3}\, \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}+4 i \sqrt {3}+\left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}+4}}{6 \left (3 \sqrt {3}+\sqrt {31}\right )^{{1}/{3}}} \\
y &= \frac {2^{{1}/{3}} \sqrt {3}\, \sqrt {\left (-4+\left (6 a^{2}-6\right ) x^{2}\right ) \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}-i \sqrt {3}\, \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}+4 i \sqrt {3}+\left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}+4}}{6 \left (3 \sqrt {3}+\sqrt {31}\right )^{{1}/{3}}} \\
y &= -\frac {2^{{1}/{3}} \sqrt {3}\, \sqrt {\left (-4+\left (6 a^{2}-6\right ) x^{2}\right ) \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}+i \sqrt {3}\, \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}-4 i \sqrt {3}+\left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}+4}}{6 \left (3 \sqrt {3}+\sqrt {31}\right )^{{1}/{3}}} \\
y &= \frac {2^{{1}/{3}} \sqrt {3}\, \sqrt {\left (-4+\left (6 a^{2}-6\right ) x^{2}\right ) \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}+i \sqrt {3}\, \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}-4 i \sqrt {3}+\left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}+4}}{6 \left (3 \sqrt {3}+\sqrt {31}\right )^{{1}/{3}}} \\
\frac {4 \left (\munderset {\textit {\_R} &=\operatorname {RootOf}\left (\textit {\_Z}^{3}+2 \textit {\_Z}^{2}+8\right )}{\sum }\frac {\ln \left (-a^{2} x^{2}+x^{2}+y^{2}-\textit {\_R} \right )}{\textit {\_R} \left (3 \textit {\_R} +4\right )}\right )+y \left (a^{2}-1\right )-c_{1} a^{4}+2 c_{1} a^{2}-c_{1}}{a^{4}-2 a^{2}+1} = 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 2.638 (sec). Leaf size: 882
DSolve[D[y[x],x] == (-8*(-1 + a)*(1 + a)*x)/(8 - 8*a^2 + 2*x^4 - 6*a^2*x^4 + 6*a^4*x^4 - 2*a^6*x^4 + x^6 - 4*a^2*x^6 + 6*a^4*x^6 - 4*a^6*x^6 + a^8*x^6 - 8*y[x] + 4*x^2*y[x]^2 - 8*a^2*x^2*y[x]^2 + 4*a^4*x^2*y[x]^2 + 3*x^4*y[x]^2 - 9*a^2*x^4*y[x]^2 + 9*a^4*x^4*y[x]^2 - 3*a^6*x^4*y[x]^2 + 2*y[x]^4 - 2*a^2*y[x]^4 + 3*x^2*y[x]^4 - 6*a^2*x^2*y[x]^4 + 3*a^4*x^2*y[x]^4 + y[x]^6 - a^2*y[x]^6),y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^x\frac {8 K[1]}{(a-1) (a+1) \left (a^6 K[1]^6-3 a^4 K[1]^6+3 a^2 K[1]^6-K[1]^6-2 a^4 K[1]^4+4 a^2 K[1]^4-3 a^4 y(x)^2 K[1]^4+6 a^2 y(x)^2 K[1]^4-3 y(x)^2 K[1]^4-2 K[1]^4+3 a^2 y(x)^4 K[1]^2-3 y(x)^4 K[1]^2+4 a^2 y(x)^2 K[1]^2-4 y(x)^2 K[1]^2-y(x)^6-2 y(x)^4-8\right )}dK[1]+\int _1^{y(x)}\left (\frac {8 K[2]}{(a-1)^2 (a+1)^2 \left (-a^6 x^6+3 a^4 x^6-3 a^2 x^6+x^6+2 a^4 x^4-4 a^2 x^4+3 a^4 K[2]^2 x^4-6 a^2 K[2]^2 x^4+3 K[2]^2 x^4+2 x^4-3 a^2 K[2]^4 x^2+3 K[2]^4 x^2-4 a^2 K[2]^2 x^2+4 K[2]^2 x^2+K[2]^6+2 K[2]^4+8\right )}-\frac {\int _1^x-\frac {8 K[1] \left (-6 K[2]^5+12 a^2 K[1]^2 K[2]^3-12 K[1]^2 K[2]^3-8 K[2]^3-6 a^4 K[1]^4 K[2]+12 a^2 K[1]^4 K[2]-6 K[1]^4 K[2]+8 a^2 K[1]^2 K[2]-8 K[1]^2 K[2]\right )}{(a-1) (a+1) \left (a^6 K[1]^6-3 a^4 K[1]^6+3 a^2 K[1]^6-K[1]^6-2 a^4 K[1]^4+4 a^2 K[1]^4-3 a^4 K[2]^2 K[1]^4+6 a^2 K[2]^2 K[1]^4-3 K[2]^2 K[1]^4-2 K[1]^4+3 a^2 K[2]^4 K[1]^2-3 K[2]^4 K[1]^2+4 a^2 K[2]^2 K[1]^2-4 K[2]^2 K[1]^2-K[2]^6-2 K[2]^4-8\right )^2}dK[1] a^2-\int _1^x-\frac {8 K[1] \left (-6 K[2]^5+12 a^2 K[1]^2 K[2]^3-12 K[1]^2 K[2]^3-8 K[2]^3-6 a^4 K[1]^4 K[2]+12 a^2 K[1]^4 K[2]-6 K[1]^4 K[2]+8 a^2 K[1]^2 K[2]-8 K[1]^2 K[2]\right )}{(a-1) (a+1) \left (a^6 K[1]^6-3 a^4 K[1]^6+3 a^2 K[1]^6-K[1]^6-2 a^4 K[1]^4+4 a^2 K[1]^4-3 a^4 K[2]^2 K[1]^4+6 a^2 K[2]^2 K[1]^4-3 K[2]^2 K[1]^4-2 K[1]^4+3 a^2 K[2]^4 K[1]^2-3 K[2]^4 K[1]^2+4 a^2 K[2]^2 K[1]^2-4 K[2]^2 K[1]^2-K[2]^6-2 K[2]^4-8\right )^2}dK[1]-1}{(a-1) (a+1)}\right )dK[2]=c_1,y(x)\right ]
\]