76.5.24 problem 25
Internal
problem
ID
[17373]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
2.
First
order
differential
equations.
Section
2.7
(Substitution
Methods).
Problems
at
page
108
Problem
number
:
25
Date
solved
:
Monday, March 31, 2025 at 04:09:44 PM
CAS
classification
:
[[_1st_order, _with_exponential_symmetries]]
\begin{align*} 1&=\left (3 \,{\mathrm e}^{y}-2 x \right ) y^{\prime } \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 364
ode:=1 = (3*exp(y(x))-2*x)*diff(y(x),x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\ln \left (2\right )-\ln \left (3\right )+\ln \left (\frac {\left (-108 c_1 +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+27 c_1^{2}}\right )^{{2}/{3}}+2 x \left (-108 c_1 +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+27 c_1^{2}}\right )^{{1}/{3}}+4 x^{2}}{\left (-108 c_1 +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+27 c_1^{2}}\right )^{{1}/{3}}}\right ) \\
y &= -2 \ln \left (2\right )-\ln \left (3\right )+\ln \left (\frac {i \left (-\left (-108 c_1 +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+27 c_1^{2}}\right )^{{2}/{3}}+4 x^{2}\right ) \sqrt {3}-{\left (\left (-108 c_1 +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+27 c_1^{2}}\right )^{{1}/{3}}-2 x \right )}^{2}}{\left (-108 c_1 +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+27 c_1^{2}}\right )^{{1}/{3}}}\right ) \\
y &= -2 \ln \left (2\right )-\ln \left (3\right )+\ln \left (\frac {i \left (\left (-108 c_1 +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+27 c_1^{2}}\right )^{{2}/{3}}-4 x^{2}\right ) \sqrt {3}-{\left (\left (-108 c_1 +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+27 c_1^{2}}\right )^{{1}/{3}}-2 x \right )}^{2}}{\left (-108 c_1 +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+27 c_1^{2}}\right )^{{1}/{3}}}\right ) \\
\end{align*}
✓ Mathematica. Time used: 0.247 (sec). Leaf size: 346
ode=1==(3*Exp[y[x]]-2*x)*D[y[x],x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \log \left (\frac {1}{3} \left (\sqrt [3]{x^3+\frac {3}{2} \left (\sqrt {3} \sqrt {c_1 \left (-4 x^3+27 c_1\right )}-9 c_1\right )}+\frac {x^2}{\sqrt [3]{x^3+\frac {3}{2} \left (\sqrt {3} \sqrt {c_1 \left (-4 x^3+27 c_1\right )}-9 c_1\right )}}+x\right )\right ) \\
y(x)\to \log \left (\frac {1}{12} \left (i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{2 x^3+3 \sqrt {3} \sqrt {c_1 \left (-4 x^3+27 c_1\right )}-27 c_1}-\frac {2 i \left (\sqrt {3}-i\right ) x^2}{\sqrt [3]{x^3+\frac {3}{2} \left (\sqrt {3} \sqrt {c_1 \left (-4 x^3+27 c_1\right )}-9 c_1\right )}}+4 x\right )\right ) \\
y(x)\to \log \left (\frac {1}{12} \left (-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{2 x^3+3 \sqrt {3} \sqrt {c_1 \left (-4 x^3+27 c_1\right )}-27 c_1}+\frac {2 i \left (\sqrt {3}+i\right ) x^2}{\sqrt [3]{x^3+\frac {3}{2} \left (\sqrt {3} \sqrt {c_1 \left (-4 x^3+27 c_1\right )}-9 c_1\right )}}+4 x\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 80.921 (sec). Leaf size: 286
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-(-2*x + 3*exp(y(x)))*Derivative(y(x), x) + 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \log {\left (- \frac {x^{2}}{3 \sqrt [3]{- \frac {27 C_{1}}{2} - x^{3} + \frac {\sqrt {- 4 x^{6} + \left (- 27 C_{1} - 2 x^{3}\right )^{2}}}{2}}} + \frac {x}{3} - \frac {\sqrt [3]{- \frac {27 C_{1}}{2} - x^{3} + \frac {\sqrt {- 4 x^{6} + \left (- 27 C_{1} - 2 x^{3}\right )^{2}}}{2}}}{3} \right )}, \ y{\left (x \right )} = \log {\left (- \frac {2 x^{2}}{3 \left (-1 - \sqrt {3} i\right ) \sqrt [3]{- \frac {27 C_{1}}{2} - x^{3} + \frac {\sqrt {- 4 x^{6} + \left (- 27 C_{1} - 2 x^{3}\right )^{2}}}{2}}} + \frac {x}{3} - \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{- \frac {27 C_{1}}{2} - x^{3} + \frac {\sqrt {- 4 x^{6} + \left (- 27 C_{1} - 2 x^{3}\right )^{2}}}{2}}}{6} \right )}, \ y{\left (x \right )} = \log {\left (- \frac {2 x^{2}}{3 \left (-1 + \sqrt {3} i\right ) \sqrt [3]{- \frac {27 C_{1}}{2} - x^{3} + \frac {\sqrt {- 4 x^{6} + \left (- 27 C_{1} - 2 x^{3}\right )^{2}}}{2}}} + \frac {x}{3} - \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{- \frac {27 C_{1}}{2} - x^{3} + \frac {\sqrt {- 4 x^{6} + \left (- 27 C_{1} - 2 x^{3}\right )^{2}}}{2}}}{6} \right )}\right ]
\]