53.3.7 problem 9

Internal problem ID [8469]
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 : 9
Date solved : Sunday, March 30, 2025 at 01:10:55 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: 0.094 (sec). Leaf size: 47

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: 0.805 (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: 7.587 (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)
 

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