34.1.6 problem 9

Internal problem ID [6030]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter 1, Nature and meaning of a differential equation between two variables. page 12
Problem number : 9
Date solved : Sunday, March 30, 2025 at 10:32:51 AM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.112 (sec). Leaf size: 91

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

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

✓ Mathematica. Time used: 0.661 (sec). Leaf size: 141

ode=(D[y[x],x]^2+1)^3==a^2*(D[y[x],{x,2}])^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 \\ 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**2*Derivative(y(x), (x, 2))**2 + (Derivative(y(x), x)**2 + 1)**3,0) 
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