32.6.23 problem Exercise 12.23, page 103

Internal problem ID [5888]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 12, Miscellaneous Methods
Problem number : Exercise 12.23, page 103
Date solved : Monday, January 27, 2025 at 01:24:41 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} x y^{\prime }&=x \,{\mathrm e}^{\frac {y}{x}}+x +y \end{align*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 20

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

\[ y \left (x \right ) = \left (\ln \left (-\frac {x}{x \,{\mathrm e}^{c_{1}}-1}\right )+c_{1} \right ) x \]

✓ Solution by Mathematica

Time used: 4.398 (sec). Leaf size: 38

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

\begin{align*} y(x)\to x \log \left (\frac {1}{2} \left (-1+\tanh \left (\frac {1}{2} (-\log (x)-c_1)\right )\right )\right ) \\ y(x)\to i \pi x \\ \end{align*}