69.11.14 problem 273

Internal problem ID [18152]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 11. Singular solutions of differential equations. Exercises page 92
Problem number : 273
Date solved : Thursday, October 02, 2025 at 03:03:36 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} 3 {y^{\prime }}^{2} x -6 y y^{\prime }+x +2 y&=0 \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 32

ode:=3*diff(y(x),x)^2*x-6*y(x)*diff(y(x),x)+x+2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= x \\ y &= -\frac {x}{3} \\ y &= \frac {4 c_1^{2}+2 c_1 x +x^{2}}{6 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.17 (sec). Leaf size: 67

ode=3*x*D[y[x],x]^2-6*y[x]*D[y[x],x]+x+2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {1}{3} x \left (-1+2 \cosh \left (-\log (x)+\sqrt {3} c_1\right )\right )\\ y(x)&\to -\frac {1}{3} x \left (-1+2 \cosh \left (\log (x)+\sqrt {3} c_1\right )\right )\\ y(x)&\to -\frac {x}{3}\\ y(x)&\to x \end{align*}

✓ Sympy. Time used: 2.352 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x*Derivative(y(x), x)**2 + x - 6*y(x)*Derivative(y(x), x) + 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {4 x^{2} e^{- C_{1}}}{3} + \frac {x}{3} + \frac {e^{C_{1}}}{12} \]