12.7.17 problem 18

Internal problem ID [1727]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Exact equations. Integrating factors. Section 2.6 Page 91
Problem number : 18
Date solved : Monday, January 27, 2025 at 05:33:21 AM
CAS classification : [_separable]

\begin{align*} a y+b x y+\left (c x +d x y\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 56

dsolve((a*y(x)+b*x*y(x))+(c*x+d*x*y(x))*diff(y(x),x)=0,y(x), singsol=all)
 

\[ y = x^{-\frac {a}{c}} {\mathrm e}^{\frac {-b x -c \operatorname {LambertW}\left (\frac {d \,x^{-\frac {a}{c}} {\mathrm e}^{\frac {-b x -c_1}{c}}}{c}\right )-c_1}{c}} \]

✓ Solution by Mathematica

Time used: 0.955 (sec). Leaf size: 42

DSolve[(a*y[x]+b*x*y[x])+(c*x+d*x*y[x])*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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