28.1.85 problem 88

Internal problem ID [4391]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 88
Date solved : Sunday, March 30, 2025 at 03:14:05 AM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} 2 x y^{\prime }-y&=y^{\prime } \ln \left (y y^{\prime }\right ) \end{align*}

✓ Maple. Time used: 0.307 (sec). Leaf size: 68

ode:=2*x*diff(y(x),x)-y(x) = diff(y(x),x)*ln(y(x)*diff(y(x),x)); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= {\mathrm e}^{x -\frac {1}{2}} \\ y &= -{\mathrm e}^{x -\frac {1}{2}} \\ y &= \sqrt {2}\, \sqrt {{\mathrm e}^{-2 x +2 c_1} \left (-c_1 +x \right )}\, {\mathrm e}^{x} \\ y &= -\sqrt {2}\, \sqrt {{\mathrm e}^{-2 x +2 c_1} \left (-c_1 +x \right )}\, {\mathrm e}^{x} \\ \end{align*}

✓ Mathematica. Time used: 0.339 (sec). Leaf size: 59

ode=2*x*D[y[x],x]-y[x]==D[y[x],x]*Log[y[x]*D[y[x],x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -e^{c_1} \sqrt {-2 x+i \pi +2 c_1} \\ y(x)\to e^{c_1} \sqrt {-2 x+i \pi +2 c_1} \\ y(x)\to 0 \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), x) - y(x) - log(y(x)*Derivative(y(x), x))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - exp(2*x + LambertW(-y(x)**2*exp(-2*x)))/y(x) cannot be solved by the factorable group method