56.14.1 problem Ex 1

Internal problem ID [14163]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IV, differential equations of the first order and higher degree than the first. Article 25. Equations solvable for \(y\). Page 52
Problem number : Ex 1
Date solved : Thursday, October 02, 2025 at 09:17:54 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

\begin{align*} 2 x y^{\prime }-y+\ln \left (y^{\prime }\right )&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 69

ode:=2*x*diff(y(x),x)-y(x)+ln(diff(y(x),x)) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -1+\sqrt {4 c_1 x +1}-\ln \left (2\right )+\ln \left (\frac {-1+\sqrt {4 c_1 x +1}}{x}\right ) \\ y &= -1-\sqrt {4 c_1 x +1}-\ln \left (2\right )+\ln \left (\frac {-1-\sqrt {4 c_1 x +1}}{x}\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.077 (sec). Leaf size: 32

ode=2*D[y[x],x]*x-y[x]+Log[D[y[x],x]]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.723 (sec). Leaf size: 29

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

\[ C_{1} - y{\left (x \right )} - \log {\left (W\left (2 x e^{y{\left (x \right )}}\right ) + 2 \right )} + W\left (2 x e^{y{\left (x \right )}}\right ) = 0 \]