23.2.353 problem 404

Internal problem ID [5708]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 404
Date solved : Tuesday, September 30, 2025 at 02:01:33 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

\begin{align*} \ln \left (y^{\prime }\right )+4 x y^{\prime }-2 y&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 75

ode:=ln(diff(y(x),x))+4*x*diff(y(x),x)-2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\ln \left (2\right )+\frac {\ln \left (\frac {-1+\sqrt {16 c_1 x +1}}{x}\right )}{2}-\frac {1}{2}+\frac {\sqrt {16 c_1 x +1}}{2} \\ y &= -\ln \left (2\right )+\frac {\ln \left (\frac {-1-\sqrt {16 c_1 x +1}}{x}\right )}{2}-\frac {1}{2}-\frac {\sqrt {16 c_1 x +1}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.088 (sec). Leaf size: 292

ode=Log[D[y[x],x]]+4*x*D[y[x],x]-2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^x\frac {W\left (4 e^{2 y(x)} K[1]\right )}{K[1] \left (W\left (4 e^{2 y(x)} K[1]\right )+2\right )}dK[1]+\int _1^{y(x)}-\frac {W\left (4 e^{2 K[2]} x\right ) \int _1^x\left (\frac {2 W\left (4 e^{2 K[2]} K[1]\right )}{K[1] \left (W\left (4 e^{2 K[2]} K[1]\right )+1\right ) \left (W\left (4 e^{2 K[2]} K[1]\right )+2\right )}-\frac {2 W\left (4 e^{2 K[2]} K[1]\right )^2}{K[1] \left (W\left (4 e^{2 K[2]} K[1]\right )+1\right ) \left (W\left (4 e^{2 K[2]} K[1]\right )+2\right )^2}\right )dK[1]+2 \int _1^x\left (\frac {2 W\left (4 e^{2 K[2]} K[1]\right )}{K[1] \left (W\left (4 e^{2 K[2]} K[1]\right )+1\right ) \left (W\left (4 e^{2 K[2]} K[1]\right )+2\right )}-\frac {2 W\left (4 e^{2 K[2]} K[1]\right )^2}{K[1] \left (W\left (4 e^{2 K[2]} K[1]\right )+1\right ) \left (W\left (4 e^{2 K[2]} K[1]\right )+2\right )^2}\right )dK[1]+4}{W\left (4 e^{2 K[2]} x\right )+2}dK[2]=c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.808 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x*Derivative(y(x), x) - 2*y(x) + log(Derivative(y(x), x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} - 2 y{\left (x \right )} - \log {\left (W\left (4 x e^{2 y{\left (x \right )}}\right ) + 2 \right )} + W\left (4 x e^{2 y{\left (x \right )}}\right ) = 0 \]