83.8.13 problem 14
Internal
problem
ID
[19068]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
II.
Equations
of
first
order
and
first
degree.
Misc
examples
on
chapter
II
at
page
25
Problem
number
:
14
Date
solved
:
Monday, March 31, 2025 at 06:43:17 PM
CAS
classification
:
[`y=_G(x,y')`]
\begin{align*} y \left (2 x^{2} y+{\mathrm e}^{x}\right )-\left ({\mathrm e}^{x}+y^{3}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 314
ode:=y(x)*(2*x^2*y(x)+exp(x))-(exp(x)+y(x)^3)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {4 x^{3}+\left (27 \,{\mathrm e}^{x}+\sqrt {-64 x^{9}-288 c_1 \,x^{6}-432 x^{3} c_1^{2}-216 c_1^{3}+729 \,{\mathrm e}^{2 x}}\right )^{{2}/{3}}+6 c_1}{3 \left (27 \,{\mathrm e}^{x}+\sqrt {-64 x^{9}-288 c_1 \,x^{6}-432 x^{3} c_1^{2}-216 c_1^{3}+729 \,{\mathrm e}^{2 x}}\right )^{{1}/{3}}} \\
y &= \frac {\left (-i \sqrt {3}-1\right ) \left (27 \,{\mathrm e}^{x}+\sqrt {-64 x^{9}-288 c_1 \,x^{6}-432 x^{3} c_1^{2}-216 c_1^{3}+729 \,{\mathrm e}^{2 x}}\right )^{{1}/{3}}}{6}+\frac {2 \left (x^{3}+\frac {3 c_1}{2}\right ) \left (i \sqrt {3}-1\right )}{3 \left (27 \,{\mathrm e}^{x}+\sqrt {-64 x^{9}-288 c_1 \,x^{6}-432 x^{3} c_1^{2}-216 c_1^{3}+729 \,{\mathrm e}^{2 x}}\right )^{{1}/{3}}} \\
y &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (27 \,{\mathrm e}^{x}+\sqrt {-64 x^{9}-288 c_1 \,x^{6}-432 x^{3} c_1^{2}-216 c_1^{3}+729 \,{\mathrm e}^{2 x}}\right )^{{2}/{3}}}{6}+\frac {2 \left (-i \sqrt {3}-1\right ) \left (x^{3}+\frac {3 c_1}{2}\right )}{3}}{\left (27 \,{\mathrm e}^{x}+\sqrt {-64 x^{9}-288 c_1 \,x^{6}-432 x^{3} c_1^{2}-216 c_1^{3}+729 \,{\mathrm e}^{2 x}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 21.616 (sec). Leaf size: 336
ode=y[x]*(2*x^2*y[x]+Exp[x])-(Exp[x]+y[x]^3)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {4 x^3+\left (27 e^x+\sqrt {729 e^{2 x}-8 \left (2 x^3+3 c_1\right ){}^3}\right ){}^{2/3}+6 c_1}{3 \sqrt [3]{27 e^x+\sqrt {729 e^{2 x}-8 \left (2 x^3+3 c_1\right ){}^3}}} \\
y(x)\to -\frac {i \left (-4 \left (\sqrt {3}+i\right ) x^3+\left (\sqrt {3}-i\right ) \left (27 e^x+\sqrt {729 e^{2 x}-8 \left (2 x^3+3 c_1\right ){}^3}\right ){}^{2/3}-6 \left (\sqrt {3}+i\right ) c_1\right )}{6 \sqrt [3]{27 e^x+\sqrt {729 e^{2 x}-8 \left (2 x^3+3 c_1\right ){}^3}}} \\
y(x)\to \frac {i \left (-4 \left (\sqrt {3}-i\right ) x^3+\left (\sqrt {3}+i\right ) \left (27 e^x+\sqrt {729 e^{2 x}-8 \left (2 x^3+3 c_1\right ){}^3}\right ){}^{2/3}-6 \left (\sqrt {3}-i\right ) c_1\right )}{6 \sqrt [3]{27 e^x+\sqrt {729 e^{2 x}-8 \left (2 x^3+3 c_1\right ){}^3}}} \\
y(x)\to 0 \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((2*x**2*y(x) + exp(x))*y(x) - (y(x)**3 + exp(x))*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -(2*x**2*y(x) + exp(x))*y(x)/(y(x)**3 + exp(x)) + Derivative(y(x), x) cannot be solved by the factorable group method