Internal problem ID [8143]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 563.
ODE order: 1.
ODE degree: -1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
Solve \begin {gather*} \boxed {\ln \left (y^{\prime }\right )+x y^{\prime }+a y+b=0} \end {gather*}
✓ Solution by Maple
Time used: 0.672 (sec). Leaf size: 66
dsolve(ln(diff(y(x),x))+x*diff(y(x),x)+a*y(x)+b=0,y(x), singsol=all)
\[ -\left ({\mathrm e}^{-a y \relax (x )-\LambertW \left (x \,{\mathrm e}^{-a y \relax (x )-b}\right )-b}\right )^{-\frac {1}{a +1}} c_{1}+x -\frac {{\mathrm e}^{a y \relax (x )+\LambertW \left (x \,{\mathrm e}^{-a y \relax (x )-b}\right )+b}}{a} = 0 \]
✓ Solution by Mathematica
Time used: 0.158 (sec). Leaf size: 59
DSolve[b + Log[y'[x]] + a*y[x] + x*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [a \left (\frac {(a+1) \log \left (1-a \text {ProductLog}\left (x e^{-a y(x)-b}\right )\right )}{a^2}+\frac {\text {ProductLog}\left (x e^{-a y(x)-b}\right )}{a}\right )+a y(x)=c_1,y(x)\right ] \]