83.31.7 problem 7

Internal problem ID [19328]
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. Exercise VII (E) at page 112
Problem number : 7
Date solved : Monday, March 31, 2025 at 07:07:07 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 1.491 (sec). Leaf size: 89

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

\begin{align*} y &= 0 \\ y &= c_{1} \\ y &= \frac {a \left ({\mathrm e}^{\frac {\sqrt {c_{1}^{2}-1}\, \left (c_2 +x \right )}{a}}-c_{1} \right )}{\sqrt {c_{1}^{2}-1}} \\ y &= c_{1} x +c_2 \\ y &= c_{1} {\mathrm e}^{-\frac {x}{a}}+c_2 \,{\mathrm e}^{\frac {x}{a}} \\ y &= c_{1} \sin \left (\frac {x}{a}\right )+c_2 \cos \left (\frac {x}{a}\right ) \\ \end{align*}

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(sqrt(4*a**2*Derivative(y(x), (x, 2))**2 + 4*y(x)*Derivative(y(x), (x, 2)) + 1)/2 + y(x)*Derivative(y(x), (x, 2)) + 1/2) + Derivative(y(x), x) cannot be solved by the factorable group method