60.2.307 problem 884
Internal
problem
ID
[10894]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
884
Date
solved
:
Tuesday, January 28, 2025 at 05:29:26 PM
CAS
classification
:
[_rational]
\begin{align*} y^{\prime }&=-\frac {\left (-1-y^{4}+2 x^{2} y^{2}-x^{4}-y^{6}+3 x^{2} y^{4}-3 x^{4} y^{2}+x^{6}\right ) x}{y} \end{align*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 112
dsolve(diff(y(x),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)*x/y(x),y(x), singsol=all)
\[
y = -{\mathrm e}^{\operatorname {RootOf}\left (-x^{2} {\mathrm e}^{2 \textit {\_Z}}+2 x^{3} {\mathrm e}^{\textit {\_Z}}+\ln \left (-\frac {-{\mathrm e}^{2 \textit {\_Z}}+2 x \,{\mathrm e}^{\textit {\_Z}}-1}{{\mathrm e}^{\textit {\_Z}}-2 x}\right ) {\mathrm e}^{2 \textit {\_Z}}-2 \ln \left (-\frac {-{\mathrm e}^{2 \textit {\_Z}}+2 x \,{\mathrm e}^{\textit {\_Z}}-1}{{\mathrm e}^{\textit {\_Z}}-2 x}\right ) {\mathrm e}^{\textit {\_Z}} x -2 c_{1} {\mathrm e}^{2 \textit {\_Z}}-\textit {\_Z} \,{\mathrm e}^{2 \textit {\_Z}}+4 c_{1} x \,{\mathrm e}^{\textit {\_Z}}+2 \textit {\_Z} x \,{\mathrm e}^{\textit {\_Z}}-1\right )}+x
\]
✓ Solution by Mathematica
Time used: 1.230 (sec). Leaf size: 810
DSolve[D[y[x],x] == (x*(1 + x^4 - x^6 - 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],y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^x\left (e^{\int _1^{(K[2]-y(x)) (K[2]+y(x))}\left (\frac {1}{1-K[1]}-\frac {2}{K[1]}\right )dK[1]} K[2]^7-e^{\int _1^{(K[2]-y(x)) (K[2]+y(x))}\left (\frac {1}{1-K[1]}-\frac {2}{K[1]}\right )dK[1]} \left (3 y(x)^2+1\right ) K[2]^5+e^{\int _1^{(K[2]-y(x)) (K[2]+y(x))}\left (\frac {1}{1-K[1]}-\frac {2}{K[1]}\right )dK[1]} y(x)^2 \left (3 y(x)^2+2\right ) K[2]^3-e^{\int _1^{(K[2]-y(x)) (K[2]+y(x))}\left (\frac {1}{1-K[1]}-\frac {2}{K[1]}\right )dK[1]} \left (y(x)^6+y(x)^4+1\right ) K[2]\right )dK[2]+\int _1^{y(x)}\left (e^{\int _1^{(x-K[3]) (x+K[3])}\left (\frac {1}{1-K[1]}-\frac {2}{K[1]}\right )dK[1]} K[3]-\int _1^x\left (-2 e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\left (\frac {1}{1-K[1]}-\frac {2}{K[1]}\right )dK[1]} K[3] \left (\frac {1}{1-(K[2]-K[3]) (K[2]+K[3])}-\frac {2}{(K[2]-K[3]) (K[2]+K[3])}\right ) K[2]^7-6 e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\left (\frac {1}{1-K[1]}-\frac {2}{K[1]}\right )dK[1]} K[3] K[2]^5+2 e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\left (\frac {1}{1-K[1]}-\frac {2}{K[1]}\right )dK[1]} K[3] \left (3 K[3]^2+1\right ) \left (\frac {1}{1-(K[2]-K[3]) (K[2]+K[3])}-\frac {2}{(K[2]-K[3]) (K[2]+K[3])}\right ) K[2]^5+6 e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\left (\frac {1}{1-K[1]}-\frac {2}{K[1]}\right )dK[1]} K[3]^3 K[2]^3+2 e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\left (\frac {1}{1-K[1]}-\frac {2}{K[1]}\right )dK[1]} K[3] \left (3 K[3]^2+2\right ) K[2]^3-2 e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\left (\frac {1}{1-K[1]}-\frac {2}{K[1]}\right )dK[1]} K[3]^3 \left (3 K[3]^2+2\right ) \left (\frac {1}{1-(K[2]-K[3]) (K[2]+K[3])}-\frac {2}{(K[2]-K[3]) (K[2]+K[3])}\right ) K[2]^3-e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\left (\frac {1}{1-K[1]}-\frac {2}{K[1]}\right )dK[1]} \left (6 K[3]^5+4 K[3]^3\right ) K[2]+2 e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\left (\frac {1}{1-K[1]}-\frac {2}{K[1]}\right )dK[1]} K[3] \left (K[3]^6+K[3]^4+1\right ) \left (\frac {1}{1-(K[2]-K[3]) (K[2]+K[3])}-\frac {2}{(K[2]-K[3]) (K[2]+K[3])}\right ) K[2]\right )dK[2]\right )dK[3]=c_1,y(x)\right ]
\]