40.5.14 problem 30

Internal problem ID [6679]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 9. Equations of first order and higher degree. Supplemetary problems. Page 65
Problem number : 30
Date solved : Sunday, March 30, 2025 at 11:17:55 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.046 (sec). Leaf size: 125

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

\begin{align*} x -\operatorname {arcsinh}\left (\frac {\sqrt {3}\, y}{3}\right )+\operatorname {arctanh}\left (\frac {3+y}{2 \sqrt {y^{2}+3}}\right )+\operatorname {arctanh}\left (\frac {-3+y}{2 \sqrt {y^{2}+3}}\right )-\ln \left (y-1\right )-\ln \left (y+1\right )-c_1 &= 0 \\ x +\operatorname {arcsinh}\left (\frac {\sqrt {3}\, y}{3}\right )-\operatorname {arctanh}\left (\frac {3+y}{2 \sqrt {y^{2}+3}}\right )-\operatorname {arctanh}\left (\frac {-3+y}{2 \sqrt {y^{2}+3}}\right )-\ln \left (y-1\right )-\ln \left (y+1\right )-c_1 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 60.327 (sec). Leaf size: 4821

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

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✓ Sympy. Time used: 1.434 (sec). Leaf size: 65

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

\[ \left [ x - 2 \log {\left (- \sqrt {y^{2}{\left (x \right )} + 3} + 2 y{\left (x \right )} \right )} - \operatorname {asinh}{\left (\frac {\sqrt {3} y{\left (x \right )}}{3} \right )} = C_{1}, \ x - 2 \log {\left (\sqrt {y^{2}{\left (x \right )} + 3} + 2 y{\left (x \right )} \right )} + \operatorname {asinh}{\left (\frac {\sqrt {3} y{\left (x \right )}}{3} \right )} = C_{1}\right ] \]