29.25.6 problem 703
Internal
problem
ID
[5293]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
25
Problem
number
:
703
Date
solved
:
Sunday, March 30, 2025 at 07:51:44 AM
CAS
classification
:
[_rational]
\begin{align*} x \left (a +b x y^{3}\right ) y^{\prime }+\left (a +c \,x^{3} y\right ) y&=0 \end{align*}
✓ Maple. Time used: 0.012 (sec). Leaf size: 488
ode:=x*(a+b*x*y(x)^3)*diff(y(x),x)+(a+c*x^3*y(x))*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {3^{{1}/{3}} \left (-b \,x^{2} \left (c \,x^{2}-2 c_1 \right ) 3^{{1}/{3}}+{\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c_1 \,c^{2} x^{6}+36 c_1^{2} c \,x^{4}-24 c_1^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{{2}/{3}}\right )}{3 {\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c_1 \,c^{2} x^{6}+36 c_1^{2} c \,x^{4}-24 c_1^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{{1}/{3}} b x} \\
y &= -\frac {3^{{1}/{3}} \left (\left (1+i \sqrt {3}\right ) {\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c_1 \,c^{2} x^{6}+36 c_1^{2} c \,x^{4}-24 c_1^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{{2}/{3}}+b \left (c \,x^{2}-2 c_1 \right ) x^{2} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right )\right )}{6 {\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c_1 \,c^{2} x^{6}+36 c_1^{2} c \,x^{4}-24 c_1^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{{1}/{3}} b x} \\
y &= \frac {3^{{1}/{3}} \left (\left (i \sqrt {3}-1\right ) {\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c_1 \,c^{2} x^{6}+36 c_1^{2} c \,x^{4}-24 c_1^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{{2}/{3}}+\left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) b \left (c \,x^{2}-2 c_1 \right ) x^{2}\right )}{6 {\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c_1 \,c^{2} x^{6}+36 c_1^{2} c \,x^{4}-24 c_1^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{{1}/{3}} b x} \\
\end{align*}
✓ Mathematica. Time used: 53.988 (sec). Leaf size: 484
ode=x(a+b x y[x]^3)D[y[x],x]+(a+c x^3 y[x])y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {x \left (-c x^2+2 c_1\right )}{\sqrt [3]{3} \sqrt [3]{9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}}}+\frac {\sqrt [3]{9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}}}{3^{2/3} b x} \\
y(x)\to \frac {i \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}\right ){}^{2/3}+\sqrt [6]{3} \left (\sqrt {3}+3 i\right ) b x^2 \left (c x^2-2 c_1\right )}{6 b x \sqrt [3]{9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}}} \\
y(x)\to \frac {\sqrt [6]{3} \left (\sqrt {3}-3 i\right ) b x^2 \left (c x^2-2 c_1\right )-i \sqrt [3]{3} \left (\sqrt {3}-i\right ) \left (9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}\right ){}^{2/3}}{6 b x \sqrt [3]{9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}}} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
ode = Eq(x*(a + b*x*y(x)**3)*Derivative(y(x), x) + (a + c*x**3*y(x))*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out