47.2.2 problem 9

Internal problem ID [9741]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 97. The p-discriminant equation. EXERCISES Page 314
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 06:32:36 PM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.182 (sec). Leaf size: 97

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

\begin{align*} y &= -\frac {1}{12 x^{2}} \\ y &= \frac {i c_1 \sqrt {3}-3 x}{3 x \,c_1^{2}} \\ y &= \frac {i c_1 \sqrt {3}-3 x}{3 x \,c_1^{2}} \\ y &= \frac {-i c_1 \sqrt {3}-3 x}{3 x \,c_1^{2}} \\ y &= \frac {-i c_1 \sqrt {3}-3 x}{3 x \,c_1^{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.308 (sec). Leaf size: 123

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

\begin{align*} \text {Solve}\left [-\frac {x \sqrt {12 x^2 y(x)+1} \text {arctanh}\left (\sqrt {12 x^2 y(x)+1}\right )}{\sqrt {12 x^4 y(x)+x^2}}-\frac {1}{2} \log (y(x))=c_1,y(x)\right ]\\ \text {Solve}\left [\frac {x \sqrt {12 x^2 y(x)+1} \text {arctanh}\left (\sqrt {12 x^2 y(x)+1}\right )}{\sqrt {12 x^4 y(x)+x^2}}-\frac {1}{2} \log (y(x))=c_1,y(x)\right ]\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 2.505 (sec). Leaf size: 10

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

\[ y{\left (x \right )} = \frac {C_{1} \left (C_{1} + \frac {2}{x}\right )}{12} \]