29.19.12 problem 525

Internal problem ID [5121]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 19
Problem number : 525
Date solved : Monday, January 27, 2025 at 10:12:19 AM
CAS classification : [_separable]

\begin{align*} x \left (a +y\right ) y^{\prime }&=y \left (B x +A \right ) \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 47

dsolve(x*(a+y(x))*diff(y(x),x) = y(x)*(B*x+A),y(x), singsol=all)
 

\[ y \left (x \right ) = x^{\frac {A}{a}} {\mathrm e}^{\frac {B x -a \operatorname {LambertW}\left (\frac {x^{\frac {A}{a}} {\mathrm e}^{\frac {B x +c_{1}}{a}}}{a}\right )+c_{1}}{a}} \]

✓ Solution by Mathematica

Time used: 1.056 (sec). Leaf size: 39

DSolve[x(a+y[x])D[y[x],x]==y[x](A+B x),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to a W\left (\frac {x^{\frac {A}{a}} e^{\frac {-a+B x+c_1}{a}}}{a}\right ) \\ y(x)\to 0 \\ \end{align*}