60.1.317 problem 323
Internal
problem
ID
[10331]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
323
Date
solved
:
Wednesday, March 05, 2025 at 10:17:49 AM
CAS
classification
:
[_rational]
\begin{align*} \left (a x y^{3}+c \right ) x y^{\prime }+\left (b \,x^{3} y+c \right ) y&=0 \end{align*}
✓ Maple. Time used: 0.008 (sec). Leaf size: 488
ode:=(a*x*y(x)^3+c)*x*diff(y(x),x)+(b*x^3*y(x)+c)*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-a \,x^{2} \left (b \,x^{2}-2 c_{1} \right ) 3^{{1}/{3}}+{\left (\left (9 c +\sqrt {\frac {3 b^{3} x^{8}-18 b^{2} c_{1} x^{6}+36 b \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a \,c^{2}}{a}}\right ) a^{2} x^{2}\right )}^{{2}/{3}}\right ) 3^{{1}/{3}}}{3 {\left (\left (9 c +\sqrt {\frac {3 b^{3} x^{8}-18 b^{2} c_{1} x^{6}+36 b \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a \,c^{2}}{a}}\right ) a^{2} x^{2}\right )}^{{1}/{3}} a x} \\
y &= -\frac {3^{{1}/{3}} \left (\left (1+i \sqrt {3}\right ) {\left (\left (9 c +\sqrt {\frac {3 b^{3} x^{8}-18 b^{2} c_{1} x^{6}+36 b \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a \,c^{2}}{a}}\right ) a^{2} x^{2}\right )}^{{2}/{3}}+\left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) x^{2} a \left (b \,x^{2}-2 c_{1} \right )\right )}{6 {\left (\left (9 c +\sqrt {\frac {3 b^{3} x^{8}-18 b^{2} c_{1} x^{6}+36 b \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a \,c^{2}}{a}}\right ) a^{2} x^{2}\right )}^{{1}/{3}} a x} \\
y &= \frac {\left (\left (i \sqrt {3}-1\right ) {\left (\left (9 c +\sqrt {\frac {3 b^{3} x^{8}-18 b^{2} c_{1} x^{6}+36 b \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a \,c^{2}}{a}}\right ) a^{2} x^{2}\right )}^{{2}/{3}}+x^{2} a \left (b \,x^{2}-2 c_{1} \right ) \left (3^{{1}/{3}}+i 3^{{5}/{6}}\right )\right ) 3^{{1}/{3}}}{6 {\left (\left (9 c +\sqrt {\frac {3 b^{3} x^{8}-18 b^{2} c_{1} x^{6}+36 b \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a \,c^{2}}{a}}\right ) a^{2} x^{2}\right )}^{{1}/{3}} a x} \\
\end{align*}
✓ Mathematica. Time used: 50.458 (sec). Leaf size: 484
ode=y[x]*(c + b*x^3*y[x]) + x*(c + a*x*y[x]^3)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {x \left (-b x^2+2 c_1\right )}{\sqrt [3]{3} \sqrt [3]{9 a^2 c x^2+\sqrt {3} \sqrt {a^3 x^4 \left (27 a c^2+x^2 \left (b x^2-2 c_1\right ){}^3\right )}}}+\frac {\sqrt [3]{9 a^2 c x^2+\sqrt {3} \sqrt {a^3 x^4 \left (27 a c^2+x^2 \left (b x^2-2 c_1\right ){}^3\right )}}}{3^{2/3} a x} \\
y(x)\to \frac {i \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (9 a^2 c x^2+\sqrt {3} \sqrt {a^3 x^4 \left (27 a c^2+x^2 \left (b x^2-2 c_1\right ){}^3\right )}\right ){}^{2/3}+\sqrt [6]{3} \left (\sqrt {3}+3 i\right ) a x^2 \left (b x^2-2 c_1\right )}{6 a x \sqrt [3]{9 a^2 c x^2+\sqrt {3} \sqrt {a^3 x^4 \left (27 a c^2+x^2 \left (b x^2-2 c_1\right ){}^3\right )}}} \\
y(x)\to \frac {\sqrt [6]{3} \left (\sqrt {3}-3 i\right ) a x^2 \left (b x^2-2 c_1\right )-i \sqrt [3]{3} \left (\sqrt {3}-i\right ) \left (9 a^2 c x^2+\sqrt {3} \sqrt {a^3 x^4 \left (27 a c^2+x^2 \left (b x^2-2 c_1\right ){}^3\right )}\right ){}^{2/3}}{6 a x \sqrt [3]{9 a^2 c x^2+\sqrt {3} \sqrt {a^3 x^4 \left (27 a c^2+x^2 \left (b 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*x*y(x)**3 + c)*Derivative(y(x), x) + (b*x**3*y(x) + c)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out