40.5.8 problem 24

Internal problem ID [6673]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 9. Equations of first order and higher degree. Supplemetary problems. Page 65
Problem number : 24
Date solved : Sunday, March 30, 2025 at 11:15:53 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational]

\begin{align*} 16 y^{3} {y^{\prime }}^{2}-4 x y^{\prime }+y&=0 \end{align*}

✓ Maple. Time used: 0.211 (sec). Leaf size: 102

ode:=16*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 &= -\frac {\sqrt {2}\, \sqrt {-x}}{2} \\ y &= \frac {\sqrt {2}\, \sqrt {-x}}{2} \\ y &= -\frac {\sqrt {2}\, \sqrt {x}}{2} \\ y &= \frac {\sqrt {2}\, \sqrt {x}}{2} \\ y &= 0 \\ y &= \operatorname {RootOf}\left (-\ln \left (x \right )-2 \int _{}^{\textit {\_Z}}\frac {4 \textit {\_a}^{4}+\sqrt {-4 \textit {\_a}^{4}+1}-1}{\textit {\_a} \left (4 \textit {\_a}^{4}-1\right )}d \textit {\_a} +c_1 \right ) \sqrt {x} \\ \end{align*}

✓ Mathematica. Time used: 0.539 (sec). Leaf size: 307

ode=16*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]{2} \sqrt [4]{e^{c_1} \left (x-2 e^{c_1}\right )} \\ y(x)\to -i \sqrt [4]{2} \sqrt [4]{e^{c_1} \left (x-2 e^{c_1}\right )} \\ y(x)\to i \sqrt [4]{2} \sqrt [4]{e^{c_1} \left (x-2 e^{c_1}\right )} \\ y(x)\to \sqrt [4]{2} \sqrt [4]{e^{c_1} \left (x-2 e^{c_1}\right )} \\ y(x)\to -\frac {\sqrt [4]{-e^{c_1} \left (-2 x+e^{c_1}\right )}}{\sqrt {2}} \\ y(x)\to -\frac {i \sqrt [4]{-e^{c_1} \left (-2 x+e^{c_1}\right )}}{\sqrt {2}} \\ y(x)\to \frac {i \sqrt [4]{-e^{c_1} \left (-2 x+e^{c_1}\right )}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt [4]{-e^{c_1} \left (-2 x+e^{c_1}\right )}}{\sqrt {2}} \\ y(x)\to 0 \\ y(x)\to -\frac {\sqrt {x}}{\sqrt {2}} \\ y(x)\to -\frac {i \sqrt {x}}{\sqrt {2}} \\ y(x)\to \frac {i \sqrt {x}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {x}}{\sqrt {2}} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x*Derivative(y(x), x) + 16*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 - 4*y(x)**4))/(8*y(x)**3) + Derivative(y(x), x) cannot be solved by the factorable group method