69.4.12 problem 57

Internal problem ID [18001]
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 : Thursday, October 02, 2025 at 02:32:55 PM
CAS classification : [_separable]

\begin{align*} {\mathrm e}^{x} \sin \left (y\right )^{3}+\left ({\mathrm e}^{2 x}+1\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: 0.972 (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: 12.036 (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)
 

\[ \left [ y{\left (x \right )} = \pi - \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {- \frac {1}{C_{1} - \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )}}}}{2} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {- \frac {1}{C_{1} - \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )}}}}{2} \right )} + \pi , \ y{\left (x \right )} = - \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {- \frac {1}{C_{1} - \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )}}}}{2} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {- \frac {1}{C_{1} - \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )}}}}{2} \right )}\right ] \]