29.37.21 problem 1143

Internal problem ID [5679]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 37
Problem number : 1143
Date solved : Sunday, March 30, 2025 at 10:00:59 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 24

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

\[ y = \operatorname {LambertW}\left (x \,{\mathrm e}^{-a}\right )+\frac {\operatorname {LambertW}\left (x \,{\mathrm e}^{-a}\right )^{2}}{2}+c_1 \]

✓ Mathematica. Time used: 0.036 (sec). Leaf size: 30

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

\[ y(x)\to \frac {1}{2} W\left (e^{-a} x\right )^2+W\left (e^{-a} x\right )+c_1 \]

✓ Sympy. Time used: 0.423 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = C_{1} + \frac {W^{2}\left (x e^{- a}\right )}{2} + W\left (x e^{- a}\right ) \]