30.6.11 problem 12
Internal
problem
ID
[7546]
Book
:
Fundamentals
of
Differential
Equations.
By
Nagle,
Saff
and
Snider.
9th
edition.
Boston.
Pearson
2018.
Section
:
Chapter
2,
First
order
differential
equations.
Review
problems.
page
79
Problem
number
:
12
Date
solved
:
Tuesday, September 30, 2025 at 04:47:06 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries]]
\begin{align*} y^{3}+4 \,{\mathrm e}^{x} y+\left (2 \,{\mathrm e}^{x}+3 y^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.078 (sec). Leaf size: 289
ode:=y(x)^3+4*exp(x)*y(x)+(2*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 (-2 \,12^{{1}/{3}}+\left (\sqrt {3}\, \sqrt {32+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 {32+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 (-2 i 2^{{2}/{3}} 3^{{5}/{6}}-i \left (\sqrt {3}\, \sqrt {32+27 \,{\mathrm e}^{-5 x} c_1^{2}}+9 \,{\mathrm e}^{-\frac {5 x}{2}} c_1 \right )^{{2}/{3}} \sqrt {3}-\left (\sqrt {3}\, \sqrt {32+27 \,{\mathrm e}^{-5 x} c_1^{2}}+9 \,{\mathrm e}^{-\frac {5 x}{2}} c_1 \right )^{{2}/{3}}+2 \,2^{{2}/{3}} 3^{{1}/{3}}\right ) {\mathrm e}^{\frac {x}{2}}}{12 \left (\sqrt {3}\, \sqrt {32+27 \,{\mathrm e}^{-5 x} c_1^{2}}+9 \,{\mathrm e}^{-\frac {5 x}{2}} c_1 \right )^{{1}/{3}}} \\
y &= \frac {{\mathrm e}^{\frac {x}{2}} \left (\left (i \sqrt {3}-1\right ) \left (\sqrt {3}\, \sqrt {32+27 \,{\mathrm e}^{-5 x} c_1^{2}}+9 \,{\mathrm e}^{-\frac {5 x}{2}} c_1 \right )^{{2}/{3}}+2 \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) 2^{{2}/{3}}\right ) 3^{{1}/{3}} 2^{{2}/{3}}}{12 \left (\sqrt {3}\, \sqrt {32+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: 57.747 (sec). Leaf size: 377
ode=(y[x]^3+4*Exp[x]*y[x] )+( 2*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 (-4 \sqrt [3]{3} e^{3 x}+\sqrt [3]{2} \left (9 c_1 e^{2 x}+\sqrt {96 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 {96 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 {96 e^{9 x}+81 c_1{}^2 e^{4 x}}}}{2 \sqrt [3]{2} 3^{2/3}}+\frac {\sqrt [3]{2} \left (\sqrt {3}+3 i\right ) e^{2 x}}{3^{5/6} \sqrt [3]{9 c_1 e^{2 x}+\sqrt {96 e^{9 x}+81 c_1{}^2 e^{4 x}}}}\\ y(x)&\to \frac {\left (-1-i \sqrt {3}\right ) e^{-x} \sqrt [3]{9 c_1 e^{2 x}+\sqrt {96 e^{9 x}+81 c_1{}^2 e^{4 x}}}}{2 \sqrt [3]{2} 3^{2/3}}+\frac {\sqrt [3]{2} \left (\sqrt {3}-3 i\right ) e^{2 x}}{3^{5/6} \sqrt [3]{9 c_1 e^{2 x}+\sqrt {96 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 + 2*exp(x))*Derivative(y(x), x) + y(x)**3 + 4*y(x)*exp(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out