60.1.428 problem 439

Internal problem ID [10442]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 439
Date solved : Sunday, March 30, 2025 at 04:43:37 PM
CAS classification : [_separable]

\begin{align*} x^{2} {y^{\prime }}^{2}+3 x y y^{\prime }+3 y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.042 (sec). Leaf size: 33

ode:=x^2*diff(y(x),x)^2+3*x*y(x)*diff(y(x),x)+3*y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= x^{-\frac {3}{2}-\frac {i \sqrt {3}}{2}} c_1 \\ y &= x^{-\frac {3}{2}+\frac {i \sqrt {3}}{2}} c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.08 (sec). Leaf size: 54

ode=3*y[x]^2 + 3*x*y[x]*D[y[x],x] + x^2*D[y[x],x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to c_1 x^{-\frac {3}{2}-\frac {i \sqrt {3}}{2}} \\ y(x)\to c_1 x^{\frac {1}{2} i \left (\sqrt {3}+3 i\right )} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.318 (sec). Leaf size: 68

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), x)**2 + 3*x*y(x)*Derivative(y(x), x) + 3*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {C_{1} \sin {\left (\frac {\sqrt {3} \log {\left (x \right )}}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} \log {\left (x \right )}}{2} \right )}}{x^{\frac {3}{2}}}, \ y{\left (x \right )} = \frac {C_{1} \sin {\left (\frac {\sqrt {3} \log {\left (x \right )}}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} \log {\left (x \right )}}{2} \right )}}{x^{\frac {3}{2}}}\right ] \]