7.17 problem 18

Internal problem ID [1077]

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.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [_separable]

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

Solution by Maple

Time used: 0.031 (sec). Leaf size: 50

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

\[ y \relax (x ) = {\mathrm e}^{-\frac {a \ln \relax (x )+b x +c \LambertW \left (\frac {d \,x^{-\frac {a}{c}} {\mathrm e}^{-\frac {b x}{c}-\frac {c_{1}}{c}}}{c}\right )+c_{1}}{c}} \]

Solution by Mathematica

Time used: 0.976 (sec). Leaf size: 42

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

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