29.16.13 problem 456

Internal problem ID [5054]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 16
Problem number : 456
Date solved : Monday, January 27, 2025 at 10:06:12 AM
CAS classification : [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Solution by Maple

Time used: 0.066 (sec). Leaf size: 63

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

\[ y \left (x \right ) = \frac {2 \operatorname {LambertW}\left (\frac {{\mathrm e}^{\frac {-c_{1} \left (b +1\right )^{2}+\left (b -1\right ) a +x \left (b +1\right )^{2}}{2 a}}}{2 a}\right ) a -b^{2} x +\left (-a -x \right ) b +a}{b +1} \]

✓ Solution by Mathematica

Time used: 5.063 (sec). Leaf size: 118

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

\begin{align*} y(x)\to \frac {2 a W\left (-e^{\frac {(b+1)^2 x}{2 a}-1+c_1}\right )+a (-b)+a-b (b+1) x}{b+1} \\ y(x)\to \frac {a (-b)+a-b (b+1) x}{b+1} \\ y(x)\to \frac {2 a W\left (-e^{\frac {(b+1)^2 x}{2 a}-1}\right )+a (-b)+a-b (b+1) x}{b+1} \\ \end{align*}