23.2.85 problem 87

Internal problem ID [5440]
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 : 87
Date solved : Tuesday, September 30, 2025 at 12:43:22 PM
CAS classification : [_quadrature]

\begin{align*} 2 {y^{\prime }}^{2}+2 \left (6 y-1\right ) y^{\prime }+3 y \left (6 y-1\right )&=0 \end{align*}

✓ Maple. Time used: 0.045 (sec). Leaf size: 59

ode:=2*diff(y(x),x)^2+2*(6*y(x)-1)*diff(y(x),x)+3*y(x)*(6*y(x)-1) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= {\frac {1}{6}} \\ y &= -\frac {\sqrt {6}\, {\mathrm e}^{-\frac {3 x}{2}+\frac {3 c_1}{2}}}{3}-{\mathrm e}^{-3 x +3 c_1} \\ y &= \frac {\sqrt {6}\, {\mathrm e}^{-\frac {3 x}{2}+\frac {3 c_1}{2}}}{3}-{\mathrm e}^{-3 x +3 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.154 (sec). Leaf size: 81

ode=2 (D[y[x],x])^2+2(6 y[x]-1)D[y[x],x]+3 y[x](6  y[x]-1)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {1}{6} e^{-3 x+3 c_1} \left (2 e^{3 x/2}+e^{3 c_1}\right )\\ y(x)&\to \frac {1}{6} e^{-3 (x+2 c_1)} \left (-1+2 e^{\frac {3 x}{2}+3 c_1}\right )\\ y(x)&\to 0\\ y(x)&\to \frac {1}{6} \end{align*}

✓ Sympy. Time used: 39.255 (sec). Leaf size: 269

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((12*y(x) - 2)*Derivative(y(x), x) + (18*y(x) - 3)*y(x) + 2*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- e^{C_{1} + 9 x}} e^{- 6 x}}{3} + \frac {e^{C_{1} - 3 x}}{3}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- e^{C_{1} + 9 x}} e^{- 6 x}}{3} + \frac {e^{C_{1} - 3 x}}{3}, \ y{\left (x \right )} = - \frac {e^{C_{1} - 3 x}}{3} - \frac {\sqrt {2} e^{- 6 x} \sqrt {e^{C_{1} + 9 x}}}{3}, \ y{\left (x \right )} = - \frac {e^{C_{1} - 3 x}}{3} + \frac {\sqrt {2} e^{- 6 x} \sqrt {e^{C_{1} + 9 x}}}{3}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- e^{C_{1} + 9 x}} e^{- 6 x}}{3} + \frac {e^{C_{1} - 3 x}}{3}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- e^{C_{1} + 9 x}} e^{- 6 x}}{3} + \frac {e^{C_{1} - 3 x}}{3}, \ y{\left (x \right )} = - \frac {e^{C_{1} - 3 x}}{3} - \frac {\sqrt {2} e^{- 6 x} \sqrt {e^{C_{1} + 9 x}}}{3}, \ y{\left (x \right )} = - \frac {e^{C_{1} - 3 x}}{3} + \frac {\sqrt {2} e^{- 6 x} \sqrt {e^{C_{1} + 9 x}}}{3}\right ] \]