83.22.4 problem 4

Internal problem ID [19179]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (E) at page 63
Problem number : 4
Date solved : Thursday, March 13, 2025 at 01:47:00 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.093 (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 \left (x \right ) &= \frac {\left (\sqrt {-x \left (-2 c_{1} +x \right )}\, \sqrt {-\frac {c_{1}^{2}}{x \left (-2 c_{1} +x \right )}}-x +c_{1} \right ) x}{\sqrt {-x \left (-2 c_{1} +x \right )}} \\ y \left (x \right ) &= \frac {\left (\sqrt {-x \left (-2 c_{1} +x \right )}\, \sqrt {-\frac {c_{1}^{2}}{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: 0.279 (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: 0.472 (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)
 

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