76.4.16 problem 20

Internal problem ID [17410]
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 2.6 (Exact equations and integrating factors). Problems at page 100
Problem number : 20
Date solved : Tuesday, January 28, 2025 at 10:06:07 AM
CAS classification : [NONE]

\begin{align*} \frac {\sin \left (y\right )}{y}-2 \,{\mathrm e}^{-x} \sin \left (x \right )+\frac {\left (\cos \left (y\right )+2 \,{\mathrm e}^{-x} \cos \left (x \right )\right ) y^{\prime }}{y}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 17

dsolve((sin(y(x))/y(x)-2*exp(-x)*sin(x)) + (cos(y(x))+2*exp(-x)*cos(x))/y(x)*diff(y(x),x)=0,y(x), singsol=all)
 

\[ {\mathrm e}^{x} \sin \left (y\right )+2 y \cos \left (x \right )+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.155 (sec). Leaf size: 73

DSolve[(Sin[y[x]]/y[x]-2*Exp[-x]*Sin[x]) + (Cos[y[x]]+2*Exp[-x]*Cos[x])/y[x]*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^x\left (e^{K[1]} \sin (y(x))-2 \sin (K[1]) y(x)\right )dK[1]+\int _1^{y(x)}\left (2 \cos (x)+e^x \cos (K[2])-\int _1^x\left (e^{K[1]} \cos (K[2])-2 \sin (K[1])\right )dK[1]\right )dK[2]=c_1,y(x)\right ] \]