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 : Tuesday, March 04, 2025 at 06:33:52 PM
CAS classification : [_quadrature]

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

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

ode:=x*(diff(y(x),x)^2-1) = 2*diff(y(x),x); 
dsolve(ode,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*}

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

ode=x*( (D[y[x],x])^2-1)==2*D[y[x],x]; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 2.061 (sec). Leaf size: 61

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

\[ \left [ y{\left (x \right )} = C_{1} - \sqrt {x^{2} + 1} + \log {\left (\sqrt {x^{2} + 1} + 1 \right )}, \ y{\left (x \right )} = C_{1} + \frac {x}{\sqrt {1 + \frac {1}{x^{2}}}} + \log {\left (x \right )} - \operatorname {asinh}{\left (\frac {1}{x} \right )} + \frac {1}{x \sqrt {1 + \frac {1}{x^{2}}}}\right ] \]