54.1.321 problem 327
Internal
problem
ID
[11635]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
327
Date
solved
:
Tuesday, September 30, 2025 at 09:51:06 PM
CAS
classification
:
[_rational]
\begin{align*} \left (x y^{4}+2 x^{2} y^{3}+2 y+x \right ) y^{\prime }+y^{5}+y&=0 \end{align*}
✓ Maple. Time used: 0.010 (sec). Leaf size: 629
ode:=(x*y(x)^4+2*x^2*y(x)^3+2*y(x)+x)*diff(y(x),x)+y(x)^5+y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {-1+\frac {\left (108 c_1^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_1^{4} x^{2}+4 c_1 \,x^{4}+18 x^{2} c_1^{2}-x^{2}-4 c_1}\, c_1 x +36 c_1 \,x^{2}-8\right )^{{1}/{3}}}{2}-\frac {2 \left (3 c_1 \,x^{2}-1\right )}{\left (108 c_1^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_1^{4} x^{2}+4 c_1 \,x^{4}+18 x^{2} c_1^{2}-x^{2}-4 c_1}\, c_1 x +36 c_1 \,x^{2}-8\right )^{{1}/{3}}}}{3 c_1 x} \\
y &= \frac {i \left (4-12 c_1 \,x^{2}-\left (108 c_1^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_1^{4} x^{2}+18 x^{2} c_1^{2}+\left (4 x^{4}-4\right ) c_1 -x^{2}}\, c_1 x +36 c_1 \,x^{2}-8\right )^{{2}/{3}}\right ) \sqrt {3}+12 c_1 \,x^{2}-{\left (\left (108 c_1^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_1^{4} x^{2}+18 x^{2} c_1^{2}+\left (4 x^{4}-4\right ) c_1 -x^{2}}\, c_1 x +36 c_1 \,x^{2}-8\right )^{{1}/{3}}+2\right )}^{2}}{12 \left (108 c_1^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_1^{4} x^{2}+18 x^{2} c_1^{2}+\left (4 x^{4}-4\right ) c_1 -x^{2}}\, c_1 x +36 c_1 \,x^{2}-8\right )^{{1}/{3}} x c_1} \\
y &= \frac {12 i \sqrt {3}\, c_1 \,x^{2}+i \left (108 c_1^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_1^{4} x^{2}+18 x^{2} c_1^{2}+\left (4 x^{4}-4\right ) c_1 -x^{2}}\, c_1 x +36 c_1 \,x^{2}-8\right )^{{2}/{3}} \sqrt {3}+12 c_1 \,x^{2}-4 i \sqrt {3}-\left (108 c_1^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_1^{4} x^{2}+18 x^{2} c_1^{2}+\left (4 x^{4}-4\right ) c_1 -x^{2}}\, c_1 x +36 c_1 \,x^{2}-8\right )^{{2}/{3}}-4 \left (108 c_1^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_1^{4} x^{2}+18 x^{2} c_1^{2}+\left (4 x^{4}-4\right ) c_1 -x^{2}}\, c_1 x +36 c_1 \,x^{2}-8\right )^{{1}/{3}}-4}{12 c_1 x \left (108 c_1^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_1^{4} x^{2}+18 x^{2} c_1^{2}+\left (4 x^{4}-4\right ) c_1 -x^{2}}\, c_1 x +36 c_1 \,x^{2}-8\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 15.622 (sec). Leaf size: 675
ode=y[x] + y[x]^5 + (x + 2*y[x] + 2*x^2*y[x]^3 + x*y[x]^4)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\frac {2 c_1 \left (3 x^2+c_1\right )}{\sqrt [3]{\frac {9}{2} \left (3+c_1{}^2\right ) x^2+\frac {3}{2} \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+c_1{}^3}}+2^{2/3} \sqrt [3]{9 \left (3+c_1{}^2\right ) x^2+3 \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+2 c_1{}^3}+2 c_1}{6 x}\\ y(x)&\to \frac {-\frac {2 i \left (\sqrt {3}-i\right ) c_1 \left (3 x^2+c_1\right )}{\sqrt [3]{\frac {9}{2} \left (3+c_1{}^2\right ) x^2+\frac {3}{2} \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+c_1{}^3}}+i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{9 \left (3+c_1{}^2\right ) x^2+3 \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+2 c_1{}^3}+4 c_1}{12 x}\\ y(x)&\to \frac {\frac {2 i \left (\sqrt {3}+i\right ) c_1 \left (3 x^2+c_1\right )}{\sqrt [3]{\frac {9}{2} \left (3+c_1{}^2\right ) x^2+\frac {3}{2} \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+c_1{}^3}}-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{9 \left (3+c_1{}^2\right ) x^2+3 \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+2 c_1{}^3}+4 c_1}{12 x}\\ y(x)&\to 0\\ y(x)&\to -\sqrt [4]{-1}\\ y(x)&\to \sqrt [4]{-1}\\ y(x)&\to -(-1)^{3/4}\\ y(x)&\to (-1)^{3/4}\\ y(x)&\to \frac {1}{2} x \left (-1+\frac {i x^2}{\sqrt {-x^4}}\right )\\ y(x)&\to -\frac {x}{2}+\frac {i \sqrt {-x^4}}{2 x} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((2*x**2*y(x)**3 + x*y(x)**4 + x + 2*y(x))*Derivative(y(x), x) + y(x)**5 + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out