82.48.6 problem Ex. 6

Internal problem ID [18916]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VIII. End of chapter problems at page 107
Problem number : Ex. 6
Date solved : Monday, March 31, 2025 at 06:24:32 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 1.696 (sec). Leaf size: 101

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

\begin{align*} y &= 0 \\ y &= c_{1} \\ y &= \frac {a n \left ({\mathrm e}^{\frac {\sqrt {c_{1}^{2}-n^{2}}\, \left (c_2 +x \right )}{a n}}-c_{1} \right )}{\sqrt {c_{1}^{2}-n^{2}}} \\ 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=D[y[x],x]^2-y[x]*D[y[x],{x,2}]==n*Sqrt[D[y[x],x]^2+a^2*D[y[x],{x,2}]^2]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-n*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(n**2/2 + n*sqrt(4*a**2*Derivative(y(x), (x, 2))**2 + n**2 + 4*y(x)*Derivative(y(x), (x, 2)))/2 + y(x)*Derivative(y(x), (x, 2))) + Derivative(y(x), x) cannot be solved by the factorable group method