53.3.6 problem 8

Internal problem ID [8468]
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 : 8
Date solved : Sunday, March 30, 2025 at 01:10:52 PM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} x^{8} {y^{\prime }}^{2}+3 x y^{\prime }+9 y&=0 \end{align*}

✓ Maple. Time used: 0.486 (sec). Leaf size: 45

ode:=x^8*diff(y(x),x)^2+3*x*diff(y(x),x)+9*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {1}{4 x^{6}} \\ y &= \frac {-x^{3}+c_1}{x^{3} c_1^{2}} \\ y &= \frac {-x^{3}-c_1}{x^{3} c_1^{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.624 (sec). Leaf size: 130

ode=x^8*(D[y[x],x])^2+3*x*D[y[x],x]+9*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [\frac {x \sqrt {4 x^6 y(x)-1} \arctan \left (\sqrt {4 x^6 y(x)-1}\right )}{3 \sqrt {x^2-4 x^8 y(x)}}-\frac {1}{6} \log (y(x))&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {\sqrt {x^2-4 x^8 y(x)} \arctan \left (\sqrt {4 x^6 y(x)-1}\right )}{3 x \sqrt {4 x^6 y(x)-1}}-\frac {1}{6} \log (y(x))&=c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 4.928 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**8*Derivative(y(x), x)**2 + 3*x*Derivative(y(x), x) + 9*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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