89.32.5 problem 5

Internal problem ID [24965]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 16. Equations of order one and higher degree. Miscellaneous Exercises at page 246
Problem number : 5
Date solved : Thursday, October 02, 2025 at 11:45:06 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} x^{6} {y^{\prime }}^{2}-2 x y^{\prime }-4 y&=0 \end{align*}

✓ Maple. Time used: 0.182 (sec). Leaf size: 89

ode:=x^6*diff(y(x),x)^2-2*x*diff(y(x),x)-4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.325 (sec). Leaf size: 128

ode=x^6*D[y[x],x]^2-2*x*D[y[x],x]-4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 2.273 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**6*Derivative(y(x), x)**2 - 2*x*Derivative(y(x), x) - 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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