85.18.10 problem 1 (j)
Internal
problem
ID
[22556]
Book
:
Applied
Differential
Equations.
By
Murray
R.
Spiegel.
3rd
edition.
1980.
Pearson.
ISBN
978-0130400970
Section
:
Chapter
two.
First
order
and
simple
higher
order
ordinary
differential
equations.
A
Exercises
at
page
52
Problem
number
:
1
(j)
Date
solved
:
Thursday, October 02, 2025 at 08:50:39 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries]]
\begin{align*} y^{3}+2 y \,{\mathrm e}^{x}+\left ({\mathrm e}^{x}+3 y^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.028 (sec). Leaf size: 287
ode:=y(x)^3+2*y(x)*exp(x)+(exp(x)+3*y(x)^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {12^{{1}/{3}} \left (12^{{1}/{3}}-\left (\sqrt {3}\, \sqrt {4+27 \,{\mathrm e}^{-5 x} c_1^{2}}+9 \,{\mathrm e}^{-\frac {5 x}{2}} c_1 \right )^{{2}/{3}}\right ) {\mathrm e}^{\frac {x}{2}}}{6 \left (\sqrt {3}\, \sqrt {4+27 \,{\mathrm e}^{-5 x} c_1^{2}}+9 \,{\mathrm e}^{-\frac {5 x}{2}} c_1 \right )^{{1}/{3}}} \\
y &= \frac {2^{{2}/{3}} 3^{{1}/{3}} \left (-i 2^{{2}/{3}} 3^{{5}/{6}}-i \left (\sqrt {3}\, \sqrt {4+27 \,{\mathrm e}^{-5 x} c_1^{2}}+9 \,{\mathrm e}^{-\frac {5 x}{2}} c_1 \right )^{{2}/{3}} \sqrt {3}+2^{{2}/{3}} 3^{{1}/{3}}-\left (\sqrt {3}\, \sqrt {4+27 \,{\mathrm e}^{-5 x} c_1^{2}}+9 \,{\mathrm e}^{-\frac {5 x}{2}} c_1 \right )^{{2}/{3}}\right ) {\mathrm e}^{\frac {x}{2}}}{12 \left (\sqrt {3}\, \sqrt {4+27 \,{\mathrm e}^{-5 x} c_1^{2}}+9 \,{\mathrm e}^{-\frac {5 x}{2}} c_1 \right )^{{1}/{3}}} \\
y &= \frac {2^{{2}/{3}} 3^{{1}/{3}} {\mathrm e}^{\frac {x}{2}} \left (\left (i \sqrt {3}-1\right ) \left (\sqrt {3}\, \sqrt {4+27 \,{\mathrm e}^{-5 x} c_1^{2}}+9 \,{\mathrm e}^{-\frac {5 x}{2}} c_1 \right )^{{2}/{3}}+\left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) 2^{{2}/{3}}\right )}{12 \left (\sqrt {3}\, \sqrt {4+27 \,{\mathrm e}^{-5 x} c_1^{2}}+9 \,{\mathrm e}^{-\frac {5 x}{2}} c_1 \right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 60.152 (sec). Leaf size: 379
ode=( y[x]^3+2*Exp[x]*y[x] )+( Exp[x]+ 3*y[x]^2 )*D[y[x],x]== 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {e^{-x} \left (-2 \sqrt [3]{3} e^{3 x}+\sqrt [3]{2} \left (9 c_1 e^{2 x}+\sqrt {12 e^{9 x}+81 c_1{}^2 e^{4 x}}\right ){}^{2/3}\right )}{6^{2/3} \sqrt [3]{9 c_1 e^{2 x}+\sqrt {12 e^{9 x}+81 c_1{}^2 e^{4 x}}}}\\ y(x)&\to \frac {i \left (\sqrt {3}+i\right ) e^{-x} \sqrt [3]{9 c_1 e^{2 x}+\sqrt {12 e^{9 x}+81 c_1{}^2 e^{4 x}}}}{2 \sqrt [3]{2} 3^{2/3}}+\frac {\left (\sqrt {3}+3 i\right ) e^{2 x}}{2^{2/3} 3^{5/6} \sqrt [3]{9 c_1 e^{2 x}+\sqrt {12 e^{9 x}+81 c_1{}^2 e^{4 x}}}}\\ y(x)&\to \frac {e^{-x} \left (2 \left (\sqrt {3}-3 i\right ) e^{3 x}+\sqrt [3]{2} \sqrt [6]{3} \left (-1-i \sqrt {3}\right ) \left (9 c_1 e^{2 x}+\sqrt {12 e^{9 x}+81 c_1{}^2 e^{4 x}}\right ){}^{2/3}\right )}{2\ 2^{2/3} 3^{5/6} \sqrt [3]{9 c_1 e^{2 x}+\sqrt {12 e^{9 x}+81 c_1{}^2 e^{4 x}}}} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((3*y(x)**2 + exp(x))*Derivative(y(x), x) + y(x)**3 + 2*y(x)*exp(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out