Internal problem ID [3829]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 37
Problem number: 1145.
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 +b y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.236 (sec). Leaf size: 66
dsolve(ln(diff(y(x),x))+x*diff(y(x),x)+a+b*y(x) = 0,y(x), singsol=all)
\[ -\left ({\mathrm e}^{-b y \relax (x )-\LambertW \left (x \,{\mathrm e}^{-b y \relax (x )-a}\right )-a}\right )^{-\frac {1}{b +1}} c_{1}+x -\frac {{\mathrm e}^{b y \relax (x )+\LambertW \left (x \,{\mathrm e}^{-b y \relax (x )-a}\right )+a}}{b} = 0 \]
✓ Solution by Mathematica
Time used: 0.155 (sec). Leaf size: 59
DSolve[Log[y'[x]]+x y'[x]+ a +b y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [b \left (\frac {(b+1) \log \left (1-b \text {ProductLog}\left (x e^{-a-b y(x)}\right )\right )}{b^2}+\frac {\text {ProductLog}\left (x e^{-a-b y(x)}\right )}{b}\right )+b y(x)=c_1,y(x)\right ] \]