53.3.2 problem 4

Internal problem ID [8464]
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 : 4
Date solved : Sunday, March 30, 2025 at 01:08:36 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} {y^{\prime }}^{2}+4 x^{5} y^{\prime }-12 x^{4} y&=0 \end{align*}

✓ Maple. Time used: 0.218 (sec). Leaf size: 23

ode:=diff(y(x),x)^2+4*x^5*diff(y(x),x)-12*x^4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.567 (sec). Leaf size: 144

ode=(D[y[x],x])^2+4*x^5*D[y[x],x]-12*x^4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [\frac {1}{6} \log (y(x))-\frac {x^2 \sqrt {x^6+3 y(x)} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6+3 y(x)}}\right )}{3 \sqrt {x^4 \left (x^6+3 y(x)\right )}}&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {x^2 \sqrt {x^6+3 y(x)} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6+3 y(x)}}\right )}{3 \sqrt {x^4 \left (x^6+3 y(x)\right )}}+\frac {1}{6} \log (y(x))&=c_1,y(x)\right ] \\ y(x)\to -\frac {x^6}{3} \\ \end{align*}

✓ Sympy. Time used: 1.843 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**5*Derivative(y(x), x) - 12*x**4*y(x) + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} \left (3 C_{1} + 2 x^{3}\right ) \]