47.2.7 problem 14

Internal problem ID [9746]
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 : 14
Date solved : Tuesday, September 30, 2025 at 06:32:39 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: 0.139 (sec). Leaf size: 307

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 &= 0 \\ \frac {\int _{\textit {\_b}}^{x}\frac {-2 \textit {\_a} +\sqrt {-y^{4}+\textit {\_a}^{2}}}{y^{4}+3 \textit {\_a}^{2}}d \textit {\_a}}{2}-\int _{}^{y}\frac {\left (1+\left (\textit {\_f}^{4}-\sqrt {-\textit {\_f}^{4}+x^{2}}\, x +x^{2}\right ) \int _{\textit {\_b}}^{x}\frac {\textit {\_f}^{4}+4 \sqrt {-\textit {\_f}^{4}+\textit {\_a}^{2}}\, \textit {\_a} -5 \textit {\_a}^{2}}{\sqrt {-\textit {\_f}^{4}+\textit {\_a}^{2}}\, \left (\textit {\_f}^{4}+3 \textit {\_a}^{2}\right )^{2}}d \textit {\_a} \right ) \textit {\_f}^{3}}{\textit {\_f}^{4}-\sqrt {-\textit {\_f}^{4}+x^{2}}\, x +x^{2}}d \textit {\_f} +c_1 &= 0 \\ -\frac {\int _{\textit {\_b}}^{x}\frac {2 \textit {\_a} +\sqrt {-y^{4}+\textit {\_a}^{2}}}{y^{4}+3 \textit {\_a}^{2}}d \textit {\_a}}{2}-\int _{}^{y}\frac {\left (1+\left (\textit {\_f}^{4}+\sqrt {-\textit {\_f}^{4}+x^{2}}\, x +x^{2}\right ) \int _{\textit {\_b}}^{x}\frac {-\textit {\_f}^{4}+5 \textit {\_a}^{2}+4 \sqrt {-\textit {\_f}^{4}+\textit {\_a}^{2}}\, \textit {\_a}}{\sqrt {-\textit {\_f}^{4}+\textit {\_a}^{2}}\, \left (\textit {\_f}^{4}+3 \textit {\_a}^{2}\right )^{2}}d \textit {\_a} \right ) \textit {\_f}^{3}}{\textit {\_f}^{4}+\sqrt {-\textit {\_f}^{4}+x^{2}}\, x +x^{2}}d \textit {\_f} +c_1 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 60.205 (sec). Leaf size: 2815

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]
 

Too large to display

✗ 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