32.1.10 problem First order with homogeneous Coefficients. Exercise 7.11, page 61

Internal problem ID [5780]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 7
Problem number : First order with homogeneous Coefficients. Exercise 7.11, page 61
Date solved : Monday, January 27, 2025 at 01:17:24 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.069 (sec). Leaf size: 63

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

\[ y \left (x \right ) = \operatorname {RootOf}\left ({\mathrm e}^{2 \textit {\_Z}} \left (4 \ln \left (x \right )^{2} {\mathrm e}^{2 \textit {\_Z}}+8 \ln \left (x \right ) {\mathrm e}^{2 \textit {\_Z}} c_{1} +4 c_{1}^{2} {\mathrm e}^{2 \textit {\_Z}}-4 \sin \left (\textit {\_Z} \right ) \ln \left (x \right ) {\mathrm e}^{\textit {\_Z}}-4 \sin \left (\textit {\_Z} \right ) {\mathrm e}^{\textit {\_Z}} c_{1} +2 \sin \left (\textit {\_Z} \right )^{2}-1\right )\right ) x \]

✓ Solution by Mathematica

Time used: 0.326 (sec). Leaf size: 39

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

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