29.37.27 problem 1149

Internal problem ID [5685]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 37
Problem number : 1149
Date solved : Tuesday, March 04, 2025 at 11:22:44 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.272 (sec). Leaf size: 17

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

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

✓ Mathematica. Time used: 0.132 (sec). Leaf size: 24

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

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

✓ Sympy. Time used: 0.449 (sec). Leaf size: 19

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

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