29.37.23 problem 1145

Internal problem ID [5681]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 37
Problem number : 1145
Date solved : Sunday, March 30, 2025 at 10:01:04 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

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

✓ Maple. Time used: 0.176 (sec). Leaf size: 54

ode:=ln(diff(y(x),x))+x*diff(y(x),x)+a+b*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ -{\left (\frac {\operatorname {LambertW}\left (x \,{\mathrm e}^{-b y-a}\right )}{x}\right )}^{-\frac {1}{b +1}} c_1 +x -\frac {x}{b \operatorname {LambertW}\left (x \,{\mathrm e}^{-b y-a}\right )} = 0 \]

✓ Mathematica. Time used: 0.139 (sec). Leaf size: 59

ode=Log[D[y[x],x]]+x*D[y[x],x]+ a +b*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [b \left (\frac {(b+1) \log \left (1-b W\left (x e^{-a-b y(x)}\right )\right )}{b^2}+\frac {W\left (x e^{-a-b y(x)}\right )}{b}\right )+b y(x)=c_1,y(x)\right ] \]

✓ Sympy. Time used: 2.967 (sec). Leaf size: 97

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(a + b*y(x) + x*Derivative(y(x), x) + log(Derivative(y(x), x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} - b \left (\begin {cases} - y{\left (x \right )} + \frac {\log {\left (x \right )}}{b} - \frac {\log {\left (W\left (e^{- a} e^{- b \left (y{\left (x \right )} - \frac {\log {\left (x \right )}}{b}\right )}\right ) - \frac {1}{b} \right )}}{b} - \frac {W\left (e^{- a} e^{- b \left (y{\left (x \right )} - \frac {\log {\left (x \right )}}{b}\right )}\right )}{b} - \frac {\log {\left (W\left (e^{- a} e^{- b \left (y{\left (x \right )} - \frac {\log {\left (x \right )}}{b}\right )}\right ) - \frac {1}{b} \right )}}{b^{2}} & \text {for}\: b \neq 0 \\- y{\left (x \right )} + \frac {\log {\left (x \right )}}{b} & \text {otherwise} \end {cases}\right ) + \log {\left (x \right )} = 0 \]