60.2.313 problem 890
Internal
problem
ID
[10900]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
890
Date
solved
:
Tuesday, January 28, 2025 at 05:30:16 PM
CAS
classification
:
[_rational]
\begin{align*} y^{\prime }&=\frac {x}{-y+1+y^{4}+2 x^{2} y^{2}+x^{4}+y^{6}+3 x^{2} y^{4}+3 x^{4} y^{2}+x^{6}} \end{align*}
✓ Solution by Maple
Time used: 0.115 (sec). Leaf size: 496
dsolve(diff(y(x),x) = x/(-y(x)+1+y(x)^4+2*x^2*y(x)^2+x^4+y(x)^6+3*x^2*y(x)^4+3*x^4*y(x)^2+x^6),y(x), singsol=all)
\begin{align*}
y &= -\frac {2^{{1}/{3}} \sqrt {3}\, \sqrt {-6 x^{2} \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}-\left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}-2 \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}-4}}{6 \left (3 \sqrt {3}+\sqrt {31}\right )^{{1}/{3}}} \\
y &= \frac {2^{{1}/{3}} \sqrt {3}\, \sqrt {-6 x^{2} \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}-\left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}-2 \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}-4}}{6 \left (3 \sqrt {3}+\sqrt {31}\right )^{{1}/{3}}} \\
y &= -\frac {2^{{5}/{6}} \sqrt {3}\, \sqrt {\left (-12 x^{2}-4\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}}{12 \left (3 \sqrt {3}+\sqrt {31}\right )^{{1}/{3}}} \\
y &= \frac {2^{{5}/{6}} \sqrt {3}\, \sqrt {\left (-12 x^{2}-4\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}}{12 \left (3 \sqrt {3}+\sqrt {31}\right )^{{1}/{3}}} \\
y &= -\frac {2^{{5}/{6}} \sqrt {3}\, \sqrt {\left (-12 x^{2}-4\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}}{12 \left (3 \sqrt {3}+\sqrt {31}\right )^{{1}/{3}}} \\
y &= \frac {2^{{5}/{6}} \sqrt {3}\, \sqrt {\left (-12 x^{2}-4\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}}{12 \left (3 \sqrt {3}+\sqrt {31}\right )^{{1}/{3}}} \\
-y+\frac {\left (\int _{}^{x^{2}+y^{2}}\frac {1}{\textit {\_a}^{3}+\textit {\_a}^{2}+1}d \textit {\_a} \right )}{2}-c_{1} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.213 (sec). Leaf size: 228
DSolve[D[y[x],x] == x/(1 + x^4 + x^6 - y[x] + 2*x^2*y[x]^2 + 3*x^4*y[x]^2 + y[x]^4 + 3*x^2*y[x]^4 + y[x]^6),y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^{y(x)}\left (-\frac {K[2]}{x^6+3 K[2]^2 x^4+x^4+3 K[2]^4 x^2+2 K[2]^2 x^2+K[2]^6+K[2]^4+1}-\int _1^x\frac {K[1] \left (6 K[2]^5+12 K[1]^2 K[2]^3+4 K[2]^3+6 K[1]^4 K[2]+4 K[1]^2 K[2]\right )}{\left (K[1]^6+3 K[2]^2 K[1]^4+K[1]^4+3 K[2]^4 K[1]^2+2 K[2]^2 K[1]^2+K[2]^6+K[2]^4+1\right )^2}dK[1]+1\right )dK[2]+\int _1^x-\frac {K[1]}{K[1]^6+3 y(x)^2 K[1]^4+K[1]^4+3 y(x)^4 K[1]^2+2 y(x)^2 K[1]^2+y(x)^6+y(x)^4+1}dK[1]=c_1,y(x)\right ]
\]