48.2.17 problem 20

Internal problem ID [9835]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 16. Nonlinear equations. Miscellaneous Exercises. Page 340
Problem number : 20
Date solved : Tuesday, September 30, 2025 at 06:42:47 PM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} x^{6} {y^{\prime }}^{2}&=16 y+8 x y^{\prime } \end{align*}

✓ Maple. Time used: 0.177 (sec). Leaf size: 89

ode:=x^6*diff(y(x),x)^2 = 16*y(x)+8*x*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.321 (sec). Leaf size: 122

ode=x^6*D[y[x],x]^2==8*(2*y[x]+x*D[y[x],x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [-\frac {x \sqrt {x^4 y(x)+1} \text {arctanh}\left (\sqrt {x^4 y(x)+1}\right )}{2 \sqrt {x^6 y(x)+x^2}}-\frac {1}{4} \log (y(x))=c_1,y(x)\right ]\\ \text {Solve}\left [\frac {x \sqrt {x^4 y(x)+1} \text {arctanh}\left (\sqrt {x^4 y(x)+1}\right )}{2 \sqrt {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: 2.402 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**6*Derivative(y(x), x)**2 - 8*x*Derivative(y(x), x) - 16*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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