60.2.319 problem 896
Internal
problem
ID
[10906]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
896
Date
solved
:
Tuesday, January 28, 2025 at 05:30:31 PM
CAS
classification
:
[_rational]
\begin{align*} y^{\prime }&=\frac {x +1+y^{4}-2 x^{2} y^{2}+x^{4}+y^{6}-3 x^{2} y^{4}+3 x^{4} y^{2}-x^{6}}{y} \end{align*}
✓ Solution by Maple
Time used: 0.022 (sec). Leaf size: 61
dsolve(diff(y(x),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)/y(x),y(x), singsol=all)
\[
-\int _{\textit {\_b}}^{y}\frac {\textit {\_a}}{\textit {\_a}^{6}-3 \textit {\_a}^{4} x^{2}+3 x^{4} \textit {\_a}^{2}-x^{6}+\textit {\_a}^{4}-2 \textit {\_a}^{2} x^{2}+x^{4}+1}d \textit {\_a} +x -c_{1} = 0
\]
✓ Solution by Mathematica
Time used: 0.235 (sec). Leaf size: 1163
DSolve[D[y[x],x] == (1 + x + 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))}\frac {(2-3 K[1]) K[1]}{(K[1]-1) K[1]^2-1}dK[1]} K[2]^6-e^{\int _1^{(K[2]-y(x)) (K[2]+y(x))}\frac {(2-3 K[1]) K[1]}{(K[1]-1) K[1]^2-1}dK[1]} \left (3 y(x)^2+1\right ) K[2]^4+e^{\int _1^{(K[2]-y(x)) (K[2]+y(x))}\frac {(2-3 K[1]) K[1]}{(K[1]-1) K[1]^2-1}dK[1]} y(x)^2 \left (3 y(x)^2+2\right ) K[2]^2-e^{\int _1^{(K[2]-y(x)) (K[2]+y(x))}\frac {(2-3 K[1]) K[1]}{(K[1]-1) K[1]^2-1}dK[1]} K[2]-e^{\int _1^{(K[2]-y(x)) (K[2]+y(x))}\frac {(2-3 K[1]) K[1]}{(K[1]-1) K[1]^2-1}dK[1]} \left (y(x)^6+y(x)^4+1\right )\right )dK[2]+\int _1^{y(x)}\left (e^{\int _1^{(x-K[3]) (x+K[3])}\frac {(2-3 K[1]) K[1]}{(K[1]-1) K[1]^2-1}dK[1]} K[3]-\int _1^x\left (-\frac {2 e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\frac {(2-3 K[1]) K[1]}{(K[1]-1) K[1]^2-1}dK[1]} (K[2]-K[3]) K[3] (K[2]+K[3]) (2-3 (K[2]-K[3]) (K[2]+K[3])) K[2]^6}{(K[2]-K[3])^2 (K[2]+K[3])^2 ((K[2]-K[3]) (K[2]+K[3])-1)-1}-6 e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\frac {(2-3 K[1]) K[1]}{(K[1]-1) K[1]^2-1}dK[1]} K[3] K[2]^4+\frac {2 e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\frac {(2-3 K[1]) K[1]}{(K[1]-1) K[1]^2-1}dK[1]} (K[2]-K[3]) K[3] (K[2]+K[3]) \left (3 K[3]^2+1\right ) (2-3 (K[2]-K[3]) (K[2]+K[3])) K[2]^4}{(K[2]-K[3])^2 (K[2]+K[3])^2 ((K[2]-K[3]) (K[2]+K[3])-1)-1}+6 e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\frac {(2-3 K[1]) K[1]}{(K[1]-1) K[1]^2-1}dK[1]} K[3]^3 K[2]^2+2 e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\frac {(2-3 K[1]) K[1]}{(K[1]-1) K[1]^2-1}dK[1]} K[3] \left (3 K[3]^2+2\right ) K[2]^2-\frac {2 e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\frac {(2-3 K[1]) K[1]}{(K[1]-1) K[1]^2-1}dK[1]} (K[2]-K[3]) K[3]^3 (K[2]+K[3]) \left (3 K[3]^2+2\right ) (2-3 (K[2]-K[3]) (K[2]+K[3])) K[2]^2}{(K[2]-K[3])^2 (K[2]+K[3])^2 ((K[2]-K[3]) (K[2]+K[3])-1)-1}+\frac {2 e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\frac {(2-3 K[1]) K[1]}{(K[1]-1) K[1]^2-1}dK[1]} (K[2]-K[3]) K[3] (K[2]+K[3]) (2-3 (K[2]-K[3]) (K[2]+K[3])) K[2]}{(K[2]-K[3])^2 (K[2]+K[3])^2 ((K[2]-K[3]) (K[2]+K[3])-1)-1}-e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\frac {(2-3 K[1]) K[1]}{(K[1]-1) K[1]^2-1}dK[1]} \left (6 K[3]^5+4 K[3]^3\right )+\frac {2 e^{\int _1^{(K[2]-K[3]) (K[2]+K[3])}\frac {(2-3 K[1]) K[1]}{(K[1]-1) K[1]^2-1}dK[1]} (K[2]-K[3]) K[3] (K[2]+K[3]) \left (K[3]^6+K[3]^4+1\right ) (2-3 (K[2]-K[3]) (K[2]+K[3]))}{(K[2]-K[3])^2 (K[2]+K[3])^2 ((K[2]-K[3]) (K[2]+K[3])-1)-1}\right )dK[2]\right )dK[3]=c_1,y(x)\right ]
\]