54.1.277 problem 283
Internal
problem
ID
[11591]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
283
Date
solved
:
Tuesday, September 30, 2025 at 09:35:59 PM
CAS
classification
:
[`y=_G(x,y')`]
\begin{align*} 3 \left (y^{2}-x^{2}\right ) y^{\prime }+2 y^{3}-6 x \left (x +1\right ) y-3 \,{\mathrm e}^{x}&=0 \end{align*}
✓ Maple. Time used: 0.009 (sec). Leaf size: 353
ode:=3*(y(x)^2-x^2)*diff(y(x),x)+2*y(x)^3-6*x*(1+x)*y(x)-3*exp(x) = 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 {\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^{{1}/{3}}}{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 {{\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 (i \sqrt {3}-1\right )}{2}+{\mathrm e}^{2 x} x^{2} \left (1+i \sqrt {3}\right )\right ) 2^{{1}/{3}}}{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: 60.172 (sec). Leaf size: 497
ode=3*(y[x]^2-x^2)*D[y[x],x]+2*y[x]^3-6*x*(x+1)*y[x]-3*Exp[x]==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