15.7.17 problem 17

Internal problem ID [2998]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 11, page 45
Problem number : 17
Date solved : Monday, January 27, 2025 at 07:07:00 AM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }+y \cos \left (x \right )&=y^{3} \sin \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 86

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

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

✓ Solution by Mathematica

Time used: 10.405 (sec). Leaf size: 84

DSolve[D[y[x],x]+y[x]*Cos[x]==y[x]^3*Sin[x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {1}{\sqrt {e^{2 \sin (x)} \left (-2 \int _1^xe^{-2 \sin (K[1])} \sin (K[1])dK[1]+c_1\right )}} \\ y(x)\to \frac {1}{\sqrt {e^{2 \sin (x)} \left (-2 \int _1^xe^{-2 \sin (K[1])} \sin (K[1])dK[1]+c_1\right )}} \\ y(x)\to 0 \\ \end{align*}