83.23.14 problem 14

Internal problem ID [19216]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter V. Singular solutions. Exercise V at page 76
Problem number : 14
Date solved : Thursday, March 13, 2025 at 01:55:50 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 32

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

\begin{align*} y \left (x \right ) &= x \\ y \left (x \right ) &= -\frac {x}{3} \\ y \left (x \right ) &= \frac {4 c_{1}^{2}+2 c_{1} x +x^{2}}{6 c_{1}} \\ \end{align*}

✓ Mathematica. Time used: 0.271 (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: 3.684 (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)
 

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