78.8.10 problem 10
Internal
problem
ID
[18137]
Book
:
DIFFERENTIAL
EQUATIONS
WITH
APPLICATIONS
AND
HISTORICAL
NOTES
by
George
F.
Simmons.
3rd
edition.
2017.
CRC
press,
Boca
Raton
FL.
Section
:
Chapter
2.
First
order
equations.
Miscellaneous
Problems
for
Chapter
2.
Problems
at
page
99
Problem
number
:
10
Date
solved
:
Monday, March 31, 2025 at 05:13:17 PM
CAS
classification
:
[_exact, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} \left ({\mathrm e}^{x}-3 x^{2} y^{2}\right ) y^{\prime }+y \,{\mathrm e}^{x}&=2 x y^{3} \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 274
ode:=(exp(x)-3*x^2*y(x)^2)*diff(y(x),x)+y(x)*exp(x) = 2*x*y(x)^3;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (108 c_1 x +12 \sqrt {81 c_1^{2} x^{2}-12 \,{\mathrm e}^{3 x}}\right )^{{2}/{3}}+12 \,{\mathrm e}^{x}}{6 \left (108 c_1 x +12 \sqrt {81 c_1^{2} x^{2}-12 \,{\mathrm e}^{3 x}}\right )^{{1}/{3}} x} \\
y &= \frac {-i \left (108 c_1 x +12 \sqrt {81 c_1^{2} x^{2}-12 \,{\mathrm e}^{3 x}}\right )^{{2}/{3}} \sqrt {3}+12 i {\mathrm e}^{x} \sqrt {3}-\left (108 c_1 x +12 \sqrt {81 c_1^{2} x^{2}-12 \,{\mathrm e}^{3 x}}\right )^{{2}/{3}}-12 \,{\mathrm e}^{x}}{12 \left (108 c_1 x +12 \sqrt {81 c_1^{2} x^{2}-12 \,{\mathrm e}^{3 x}}\right )^{{1}/{3}} x} \\
y &= -\frac {-i \left (108 c_1 x +12 \sqrt {81 c_1^{2} x^{2}-12 \,{\mathrm e}^{3 x}}\right )^{{2}/{3}} \sqrt {3}+12 i {\mathrm e}^{x} \sqrt {3}+\left (108 c_1 x +12 \sqrt {81 c_1^{2} x^{2}-12 \,{\mathrm e}^{3 x}}\right )^{{2}/{3}}+12 \,{\mathrm e}^{x}}{12 \left (108 c_1 x +12 \sqrt {81 c_1^{2} x^{2}-12 \,{\mathrm e}^{3 x}}\right )^{{1}/{3}} x} \\
\end{align*}
✓ Mathematica. Time used: 60.255 (sec). Leaf size: 364
ode=(Exp[x]-3*x^2*y[x]^2)*D[y[x],x]+y[x]*Exp[x]==2*x*y[x]^3;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {2 \sqrt [3]{3} e^x x^2+\sqrt [3]{2} \left (9 c_1 x^4+\sqrt {-12 e^{3 x} x^6+81 c_1{}^2 x^8}\right ){}^{2/3}}{6^{2/3} x^2 \sqrt [3]{9 c_1 x^4+\sqrt {-12 e^{3 x} x^6+81 c_1{}^2 x^8}}} \\
y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{9 c_1 x^4+\sqrt {-12 e^{3 x} x^6+81 c_1{}^2 x^8}}}{2 \sqrt [3]{2} 3^{2/3} x^2}-\frac {\left (\sqrt {3}+3 i\right ) e^x}{2^{2/3} 3^{5/6} \sqrt [3]{9 c_1 x^4+\sqrt {-12 e^{3 x} x^6+81 c_1{}^2 x^8}}} \\
y(x)\to \frac {\left (-1-i \sqrt {3}\right ) \sqrt [3]{9 c_1 x^4+\sqrt {-12 e^{3 x} x^6+81 c_1{}^2 x^8}}}{2 \sqrt [3]{2} 3^{2/3} x^2}-\frac {\left (\sqrt {3}-3 i\right ) e^x}{2^{2/3} 3^{5/6} \sqrt [3]{9 c_1 x^4+\sqrt {-12 e^{3 x} x^6+81 c_1{}^2 x^8}}} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-2*x*y(x)**3 + (-3*x**2*y(x)**2 + exp(x))*Derivative(y(x), x) + y(x)*exp(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out