28.1.79 problem 82

Internal problem ID [4385]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 82
Date solved : Monday, January 27, 2025 at 09:09:47 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.031 (sec). Leaf size: 49

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

\begin{align*} y \left (x \right ) &= \sqrt {x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right )+\ln \left (x \right )+c_{1} \\ y \left (x \right ) &= -\sqrt {x^{2}+1}+\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right )+\ln \left (x \right )+c_{1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.034 (sec). Leaf size: 59

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

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