29.37.7 problem 1120

Internal problem ID [5665]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 37
Problem number : 1120
Date solved : Sunday, March 30, 2025 at 09:54:19 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.050 (sec). Leaf size: 33

ode:=(1+diff(y(x),x)^2)^(1/2) = x*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 41

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

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

✓ Sympy. Time used: 1.099 (sec). Leaf size: 41

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

\[ \left [ y{\left (x \right )} = C_{1} - \begin {cases} \log {\left (x + \sqrt {x^{2} - 1} \right )} & \text {for}\: x > -1 \wedge x < 1 \end {cases}, \ y{\left (x \right )} = C_{1} + \begin {cases} \log {\left (x + \sqrt {x^{2} - 1} \right )} & \text {for}\: x > -1 \wedge x < 1 \end {cases}\right ] \]