53.2.6 problem 13

Internal problem ID [8459]
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 : 13
Date solved : Sunday, March 30, 2025 at 01:06:29 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational]

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

✓ Maple. Time used: 0.230 (sec). Leaf size: 85

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

\begin{align*} y &= \sqrt {-x} \\ y &= -\sqrt {-x} \\ y &= \sqrt {x} \\ y &= -\sqrt {x} \\ y &= 0 \\ y &= \operatorname {RootOf}\left (-\ln \left (x \right )+2 \int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{4}-\sqrt {-\textit {\_a}^{4}+1}-1}{\textit {\_a} \left (\textit {\_a}^{4}-1\right )}d \textit {\_a} +c_1 \right ) \sqrt {x} \\ \end{align*}

✓ Mathematica. Time used: 0.382 (sec). Leaf size: 150

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

\begin{align*} y(x)\to -\sqrt [4]{-e^{c_1} \left (-2 x+e^{c_1}\right )} \\ y(x)\to -i \sqrt [4]{-e^{c_1} \left (-2 x+e^{c_1}\right )} \\ y(x)\to i \sqrt [4]{-e^{c_1} \left (-2 x+e^{c_1}\right )} \\ y(x)\to \sqrt [4]{-e^{c_1} \left (-2 x+e^{c_1}\right )} \\ y(x)\to 0 \\ y(x)\to -\sqrt {x} \\ y(x)\to -i \sqrt {x} \\ y(x)\to i \sqrt {x} \\ y(x)\to \sqrt {x} \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -(x + sqrt(x**2 - y(x)**4))/(2*y(x)**3) + Derivative(y(x), x) cannot be solved by the factorable group method