31.6.14 problem 14

Internal problem ID [5763]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 7
Problem number : 14
Date solved : Sunday, March 30, 2025 at 10:08:35 AM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.204 (sec). Leaf size: 97

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

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

✓ Mathematica. Time used: 0.274 (sec). Leaf size: 37

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

\begin{align*} y(x)\to -\sqrt {-x (x-c_1)} \\ y(x)\to \sqrt {-x (x-c_1)} \\ \end{align*}

✓ Sympy. Time used: 0.477 (sec). Leaf size: 22

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

\[ \left [ y{\left (x \right )} = - \sqrt {x \left (C_{1} - x\right )}, \ y{\left (x \right )} = \sqrt {x \left (C_{1} - x\right )}\right ] \]