47.2.57 problem 53

Internal problem ID [7473]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems. page 12
Problem number : 53
Date solved : Sunday, March 30, 2025 at 12:09:51 PM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} y \left (1+\sqrt {x^{2} y^{4}-1}\right )+2 x y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 32

ode:=y(x)*(1+(x^2*y(x)^4-1)^(1/2))+2*x*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+c_1 -2 \int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a} \sqrt {\textit {\_a}^{4}-1}}d \textit {\_a} \right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 3.235 (sec). Leaf size: 109

ode=y[x]*(1+Sqrt[x^2*y[x]^4-1])+2*x*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {\sqrt [4]{\sec ^2(-\log (x)+c_1)}}{\sqrt {x}} \\ y(x)\to -\frac {i \sqrt [4]{\sec ^2(-\log (x)+c_1)}}{\sqrt {x}} \\ y(x)\to \frac {i \sqrt [4]{\sec ^2(-\log (x)+c_1)}}{\sqrt {x}} \\ y(x)\to \frac {\sqrt [4]{\sec ^2(-\log (x)+c_1)}}{\sqrt {x}} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.776 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), x) + (sqrt(x**2*y(x)**4 - 1) + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \begin {cases} - i \operatorname {acosh}{\left (\frac {1}{x y^{2}{\left (x \right )}} \right )} & \text {for}\: \frac {1}{\left |{x^{2} y^{4}{\left (x \right )}}\right |} > 1 \\\operatorname {asin}{\left (\frac {1}{x y^{2}{\left (x \right )}} \right )} & \text {otherwise} \end {cases} = C_{1} + \log {\left (x \right )} \]