29.7.9 problem 6

Internal problem ID [7326]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 8, Ordinary differential equations. Section 7. Other second-Order equations. page 435
Problem number : 6
Date solved : Tuesday, September 30, 2025 at 04:28:57 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear]]

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

✓ Maple. Time used: 0.044 (sec). Leaf size: 19

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

\[ y = -x -\frac {4}{k^{2} \left (c_1 +x \right )}+c_2 \]

✓ Mathematica. Time used: 0.357 (sec). Leaf size: 75

ode=k==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-\frac {i \sqrt {k^2 x^2+2 c_1 k x-1+c_1{}^2}}{k}\\ y(x)&\to \frac {i \sqrt {k^2 x^2+2 c_1 k x-1+c_1{}^2}}{k}+c_2 \end{align*}

✗ Sympy

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

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