47.2.20 problem 20

Internal problem ID [7436]
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 : 20
Date solved : Monday, January 27, 2025 at 02:57:42 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 28

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

\[ \frac {-c_{1} x^{2}+y+\sqrt {y^{2}-x^{2}}}{x^{2}} = 0 \]

✓ Solution by Mathematica

Time used: 0.391 (sec). Leaf size: 14

DSolve[x*D[y[x],x]==y[x]+Sqrt[y[x]^2-x^2],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -x \cosh (\log (x)+c_1) \]