75.4.12 problem 57

Internal problem ID [16714]
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 : Tuesday, January 28, 2025 at 09:19:15 AM
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*}

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 39

dsolve(exp(x)*sin(y(x))^3+(1+exp(2*x))*cos(y(x))*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y &= \arctan \left (\frac {\sqrt {2}\, \sqrt {\frac {1}{\arctan \left ({\mathrm e}^{x}\right )+c_{1}}}}{2}\right ) \\ y &= -\arctan \left (\frac {\sqrt {2}\, \sqrt {\frac {1}{\arctan \left ({\mathrm e}^{x}\right )+c_{1}}}}{2}\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.714 (sec). Leaf size: 56

DSolve[Exp[x]*Sin[y[x]]^3+(1+Exp[2*x])*Cos[y[x]]*D[y[x],x]==0,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*}