47.1.6 problem 6

Internal problem ID [7387]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.1 Separable equations problems. page 7
Problem number : 6
Date solved : Monday, January 27, 2025 at 02:51:36 PM
CAS classification : [_separable]

\begin{align*} x y y^{\prime }&=\sqrt {1+y^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 17

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

\[ \ln \left (x \right )-\sqrt {1+y^{2}}+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.240 (sec). Leaf size: 65

DSolve[x*y[x]*D[y[x],x]==Sqrt[1+y[x]^2],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\sqrt {\log ^2(x)+2 c_1 \log (x)-1+c_1{}^2} \\ y(x)\to \sqrt {\log ^2(x)+2 c_1 \log (x)-1+c_1{}^2} \\ y(x)\to -i \\ y(x)\to i \\ \end{align*}