47.3.20 problem 23

Internal problem ID [9768]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 99. Clairaut equation. EXERCISES Page 320
Problem number : 23
Date solved : Tuesday, September 30, 2025 at 06:40:56 PM
CAS classification : [[_homogeneous, `class G`], _rational, _dAlembert]

\begin{align*} 4 x {y^{\prime }}^{2}-3 y y^{\prime }+3&=0 \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 123

ode:=4*x*diff(y(x),x)^2-3*y(x)*diff(y(x),x)+3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {2 x \left (6+\sqrt {16 c_1 x +9}\right )}{3 \sqrt {x \left (3+\sqrt {16 c_1 x +9}\right )}} \\ y &= \frac {2 x \left (6+\sqrt {16 c_1 x +9}\right )}{3 \sqrt {x \left (3+\sqrt {16 c_1 x +9}\right )}} \\ y &= \frac {2 x \left (-6+\sqrt {16 c_1 x +9}\right )}{3 \sqrt {-x \left (-3+\sqrt {16 c_1 x +9}\right )}} \\ y &= -\frac {2 x \left (-6+\sqrt {16 c_1 x +9}\right )}{3 \sqrt {-x \left (-3+\sqrt {16 c_1 x +9}\right )}} \\ \end{align*}

✓ Mathematica. Time used: 18.9 (sec). Leaf size: 187

ode=4*x*(D[y[x],x])^2-3*y[x]*D[y[x],x]+3==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {\sqrt {432 x-e^{-\frac {c_1}{2}} \left (-144 x+e^{c_1}\right ){}^{3/2}+e^{c_1}}}{6 \sqrt {3}}\\ y(x)&\to \frac {\sqrt {432 x-e^{-\frac {c_1}{2}} \left (-144 x+e^{c_1}\right ){}^{3/2}+e^{c_1}}}{6 \sqrt {3}}\\ y(x)&\to -\frac {\sqrt {432 x+e^{-\frac {c_1}{2}} \left (-144 x+e^{c_1}\right ){}^{3/2}+e^{c_1}}}{6 \sqrt {3}}\\ y(x)&\to \frac {\sqrt {432 x+e^{-\frac {c_1}{2}} \left (-144 x+e^{c_1}\right ){}^{3/2}+e^{c_1}}}{6 \sqrt {3}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x*Derivative(y(x), x)**2 - 3*y(x)*Derivative(y(x), x) + 3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt(-48*x + 9*y(x)**2)/8 + 3*y(x)/8)/x c