29.22.20 problem 628
Internal
problem
ID
[5220]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
22
Problem
number
:
628
Date
solved
:
Sunday, March 30, 2025 at 06:54:50 AM
CAS
classification
:
[`y=_G(x,y')`]
\begin{align*} 3 \left (x^{2}-y^{2}\right ) y^{\prime }+3 \,{\mathrm e}^{x}+6 x y \left (1+x \right )-2 y^{3}&=0 \end{align*}
✓ Maple. Time used: 0.008 (sec). Leaf size: 341
ode:=3*(x^2-y(x)^2)*diff(y(x),x)+3*exp(x)+6*x*y(x)*(1+x)-2*y(x)^3 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {2^{{1}/{3}} \left (2 x^{2} {\mathrm e}^{4 x}+2^{{1}/{3}} {\left (\left ({\mathrm e}^{3 x}+c_1 +\sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 \,{\mathrm e}^{3 x} c_1 +c_1^{2}}\right ) {\mathrm e}^{4 x}\right )}^{{2}/{3}}\right ) {\mathrm e}^{-2 x}}{2 {\left (\left ({\mathrm e}^{3 x}+c_1 +\sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 \,{\mathrm e}^{3 x} c_1 +c_1^{2}}\right ) {\mathrm e}^{4 x}\right )}^{{1}/{3}}} \\
y &= \frac {2^{{1}/{3}} \left (-\frac {{\mathrm e}^{-2 x} 2^{{1}/{3}} {\left (\left ({\mathrm e}^{3 x}+c_1 +\sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 \,{\mathrm e}^{3 x} c_1 +c_1^{2}}\right ) {\mathrm e}^{4 x}\right )}^{{2}/{3}} \left (1+i \sqrt {3}\right )}{2}+{\mathrm e}^{2 x} x^{2} \left (i \sqrt {3}-1\right )\right )}{2 {\left (\left ({\mathrm e}^{3 x}+c_1 +\sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 \,{\mathrm e}^{3 x} c_1 +c_1^{2}}\right ) {\mathrm e}^{4 x}\right )}^{{1}/{3}}} \\
y &= -\frac {\left (-\frac {2^{{1}/{3}} {\left (\left ({\mathrm e}^{3 x}+c_1 +\sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 \,{\mathrm e}^{3 x} c_1 +c_1^{2}}\right ) {\mathrm e}^{4 x}\right )}^{{2}/{3}} \left (i \sqrt {3}-1\right )}{2}+x^{2} \left (1+i \sqrt {3}\right ) {\mathrm e}^{4 x}\right ) 2^{{1}/{3}} {\mathrm e}^{-2 x}}{2 {\left (\left ({\mathrm e}^{3 x}+c_1 +\sqrt {-4 x^{6} {\mathrm e}^{4 x}+{\mathrm e}^{6 x}+2 \,{\mathrm e}^{3 x} c_1 +c_1^{2}}\right ) {\mathrm e}^{4 x}\right )}^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 60.3 (sec). Leaf size: 497
ode=3(x^2-y[x]^2)D[y[x],x]+3 Exp[x]+6 x y[x](1+x)-2 y[x]^3==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {e^{-2 x} \sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} e^{2 x} x^2}{\sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) e^{-2 x} \sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}}{2 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) e^{2 x} x^2}{2^{2/3} \sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}} \\
y(x)\to \frac {\left (1+i \sqrt {3}\right ) e^{-2 x} \sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}}{2 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) e^{2 x} x^2}{2^{2/3} \sqrt [3]{\sqrt {e^{8 x} \left (-4 e^{4 x} x^6+e^{6 x}-2 c_1 e^{3 x}+c_1{}^2\right )}-e^{7 x}+c_1 e^{4 x}}} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(6*x*(x + 1)*y(x) + (3*x**2 - 3*y(x)**2)*Derivative(y(x), x) - 2*y(x)**3 + 3*exp(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (-2*x**2*y(x) - 2*x*y(x) + 2*y(x)**3/3 - exp(x))/(x**2 - y(x)**2) cannot be solved by the factorable group method