89.30.7 problem 9

Internal problem ID [24924]
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. Exercises at page 243
Problem number : 9
Date solved : Thursday, October 02, 2025 at 10:49:46 PM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 49

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

\begin{align*} y &= -\frac {2 i}{x} \\ y &= \frac {2 i}{x} \\ y &= \frac {2 \sinh \left (-\ln \left (x \right )+c_1 \right )}{x} \\ y &= -\frac {2 \sinh \left (-\ln \left (x \right )+c_1 \right )}{x} \\ \end{align*}

✓ Mathematica. Time used: 0.469 (sec). Leaf size: 71

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

\begin{align*} y(x)&\to \frac {4 e^{c_1}}{x^2}-\frac {e^{-c_1}}{4}\\ y(x)&\to \frac {e^{-c_1}}{4}-\frac {4 e^{c_1}}{x^2}\\ y(x)&\to -\frac {2 i}{x}\\ y(x)&\to \frac {2 i}{x} \end{align*}

✓ Sympy. Time used: 4.914 (sec). Leaf size: 26

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

\[ \left [ y{\left (x \right )} = - \frac {2 \sinh {\left (C_{1} + \log {\left (x \right )} \right )}}{x}, \ y{\left (x \right )} = \frac {2 \sinh {\left (C_{1} + \log {\left (x \right )} \right )}}{x}\right ] \]