60.2.325 problem 902
Internal
problem
ID
[10912]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
902
Date
solved
:
Tuesday, January 28, 2025 at 05:30:48 PM
CAS
classification
:
[_rational]
\begin{align*} y^{\prime }&=\frac {-x y^{2}+x^{3}-x -y^{6}+3 x^{2} y^{4}-3 x^{4} y^{2}+x^{6}}{\left (-y^{2}+x^{2}-1\right ) y} \end{align*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 177
dsolve(diff(y(x),x) = (-x*y(x)^2+x^3-x-y(x)^6+3*x^2*y(x)^4-3*x^4*y(x)^2+x^6)/(-y(x)^2+x^2-1)/y(x),y(x), singsol=all)
\begin{align*}
y &= \frac {\sqrt {\left (-x +c_{1} \right ) \left (4 c_{1} x^{2}-4 x^{3}+\sqrt {4 c_{1} -4 x +1}+1\right )}}{-2 c_{1} +2 x} \\
y &= \frac {\sqrt {\left (-x +c_{1} \right ) \left (4 c_{1} x^{2}-4 x^{3}+\sqrt {4 c_{1} -4 x +1}+1\right )}}{-2 x +2 c_{1}} \\
y &= \frac {\sqrt {\left (-4 c_{1} x^{2}+4 x^{3}+\sqrt {4 c_{1} -4 x +1}-1\right ) \left (x -c_{1} \right )}}{-2 c_{1} +2 x} \\
y &= \frac {\sqrt {\left (-4 c_{1} x^{2}+4 x^{3}+\sqrt {4 c_{1} -4 x +1}-1\right ) \left (x -c_{1} \right )}}{-2 x +2 c_{1}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.481 (sec). Leaf size: 377
DSolve[D[y[x],x] == (-x + x^3 + x^6 - x*y[x]^2 - 3*x^4*y[x]^2 + 3*x^2*y[x]^4 - y[x]^6)/(y[x]*(-1 + x^2 - y[x]^2)),y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^{y(x)}\left (\frac {1-4 x^2}{16 x^3 (K[2]-x)^2}-\int _1^x\left (-\frac {3 \left (-4 K[2]^2-1\right )}{16 K[2]^4 (K[1]+K[2])^2}-\frac {-4 K[2]^2-1}{8 K[2]^3 (K[1]+K[2])^3}+\frac {4 K[2]^2+1}{8 (K[1]-K[2])^3 K[2]^3}-\frac {3 \left (4 K[2]^2+1\right )}{16 (K[1]-K[2])^2 K[2]^4}+\frac {1}{2 (K[1]-K[2])^2 K[2]^2}-\frac {3}{8 (K[1]-K[2])^4 K[2]^2}-\frac {1}{2 K[2]^2 (K[1]+K[2])^2}+\frac {1}{4 (K[1]-K[2])^3 K[2]^3}+\frac {1}{4 K[2]^3 (K[1]+K[2])^3}+\frac {3}{8 K[2]^2 (K[1]+K[2])^4}\right )dK[1]+\frac {4 x^2-1}{16 x^3 (x+K[2])^2}-\frac {1}{8 x^2 (K[2]-x)^3}-\frac {1}{8 x^2 (x+K[2])^3}\right )dK[2]+\int _1^x\left (\frac {-4 y(x)^2-1}{16 y(x)^3 (K[1]+y(x))^2}+\frac {4 y(x)^2+1}{16 (K[1]-y(x))^2 y(x)^3}-\frac {1}{8 (K[1]-y(x))^3 y(x)^2}-\frac {1}{8 y(x)^2 (K[1]+y(x))^3}+1\right )dK[1]=c_1,y(x)\right ]
\]