78.1.18 problem 11.b

Internal problem ID [20944]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 1, First order ODEs. Problems section 1.5
Problem number : 11.b
Date solved : Thursday, October 02, 2025 at 06:49:53 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 69

ode:=y(x) = 2*x*diff(y(x),x)+ln(diff(y(x),x)); 
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: 0.056 (sec). Leaf size: 32

ode=y[x]==2*x*D[y[x],x]+Log[D[y[x],x]]; 
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)
 

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