54.1.278 problem 284

Internal problem ID [11592]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 284
Date solved : Tuesday, September 30, 2025 at 09:36:01 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 32

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

\[ y = \frac {{\mathrm e}^{-c_1} \sqrt {\frac {{\mathrm e}^{2 c_1} x^{2}}{\operatorname {LambertW}\left (\frac {{\mathrm e}^{2 c_1} x^{2}}{4}\right )}}}{2} \]

✓ Mathematica. Time used: 7.649 (sec). Leaf size: 64

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

\begin{align*} y(x)&\to -\frac {x}{2 \sqrt {W\left (\frac {1}{4} e^{-\frac {c_1}{2}} x^2\right )}}\\ y(x)&\to \frac {x}{2 \sqrt {W\left (\frac {1}{4} e^{-\frac {c_1}{2}} x^2\right )}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.787 (sec). Leaf size: 19

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

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