83.35.8 problem 8

Internal problem ID [19349]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Misc. Exercise on chapter VII. Page 118
Problem number : 8
Date solved : Thursday, March 13, 2025 at 02:17:10 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} -a y^{\prime \prime }&=\left (1+{y^{\prime }}^{2}\right )^{{3}/{2}} \end{align*}

✓ Maple. Time used: 0.069 (sec). Leaf size: 91

ode:=-a*diff(diff(y(x),x),x) = (1+diff(y(x),x)^2)^(3/2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y \left (x \right ) &= -i x +c_{1} \\ y \left (x \right ) &= i x +c_{1} \\ y \left (x \right ) &= \frac {\left (a +x +c_{1} \right ) \left (-a +x +c_{1} \right )}{\sqrt {a^{2}-c_{1}^{2}-2 c_{1} x -x^{2}}}+c_{2} \\ y \left (x \right ) &= \frac {\left (a +x +c_{1} \right ) \left (a -x -c_{1} \right )}{\sqrt {a^{2}-c_{1}^{2}-2 c_{1} x -x^{2}}}+c_{2} \\ \end{align*}

✓ Mathematica. Time used: 0.591 (sec). Leaf size: 71

ode=-a*D[y[x],{x,2}]==(1+D[y[x],x]^2)^(3/2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to c_2-i \sqrt {a^2 \left (-1+c_1{}^2\right )-2 a c_1 x+x^2} \\ y(x)\to i \sqrt {a^2 \left (-1+c_1{}^2\right )-2 a c_1 x+x^2}+c_2 \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*Derivative(y(x), (x, 2)) - (Derivative(y(x), x)**2 + 1)**(3/2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(-(a**2*Derivative(y(x), (x, 2))**2)**(1/3)/2 + sqrt(3)*I*(a**2*Derivative(y(x), (x, 2))**2)**(1/3)/2 - 1) + Derivative(y(x), x) cannot be solved by the factorable group method