75.4.12 problem 57
Internal
problem
ID
[16643]
Book
:
A
book
of
problems
in
ordinary
differential
equations.
M.L.
KRASNOV,
A.L.
KISELYOV,
G.I.
MARKARENKO.
MIR,
MOSCOW.
1983
Section
:
Section
4.
Equations
with
variables
separable
and
equations
reducible
to
them.
Exercises
page
38
Problem
number
:
57
Date
solved
:
Monday, March 31, 2025 at 03:03:03 PM
CAS
classification
:
[_separable]
\begin{align*} {\mathrm e}^{x} \sin \left (y\right )^{3}+\left (1+{\mathrm e}^{2 x}\right ) \cos \left (y\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.008 (sec). Leaf size: 33
ode:=exp(x)*sin(y(x))^3+(1+exp(2*x))*cos(y(x))*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \arctan \left (\sqrt {\frac {1}{2 c_1 +\arctan \left (\sinh \left (x \right )\right )}}\right ) \\
y &= -\arctan \left (\sqrt {\frac {1}{2 c_1 +\arctan \left (\sinh \left (x \right )\right )}}\right ) \\
\end{align*}
✓ Mathematica. Time used: 1.714 (sec). Leaf size: 56
ode=Exp[x]*Sin[y[x]]^3+(1+Exp[2*x])*Cos[y[x]]*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\csc ^{-1}\left (\sqrt {2} \sqrt {\arctan \left (e^x\right )-4 c_1}\right ) \\
y(x)\to \csc ^{-1}\left (\sqrt {2} \sqrt {\arctan \left (e^x\right )-4 c_1}\right ) \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 18.697 (sec). Leaf size: 107
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((exp(2*x) + 1)*cos(y(x))*Derivative(y(x), x) + exp(x)*sin(y(x))**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \pi - \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {- \frac {1}{C_{1} - \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )}}}}{2} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {- \frac {1}{C_{1} - \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )}}}}{2} \right )} + \pi , \ y{\left (x \right )} = - \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {- \frac {1}{C_{1} - \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )}}}}{2} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {- \frac {1}{C_{1} - \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )}}}}{2} \right )}\right ]
\]