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 : Monday, January 27, 2025 at 03:01:31 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*}

✓ Solution by Maple

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

dsolve(y(x)*(1+sqrt(x^2*y(x)^4-1))+2*x*diff(y(x),x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

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

DSolve[y[x]*(1+Sqrt[x^2*y[x]^4-1])+2*x*D[y[x],x]==0,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*}