77.1.83 problem 109 (page 162)

Internal problem ID [17894]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 109 (page 162)
Date solved : Thursday, March 13, 2025 at 11:09:19 AM
CAS classification : [[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]

\begin{align*} y^{\prime \prime \prime }&=\sqrt {1+{y^{\prime \prime }}^{2}} \end{align*}

✓ Maple. Time used: 2.017 (sec). Leaf size: 39

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

\begin{align*} y &= -\frac {1}{2} i x^{2}+c_{1} x +c_{2} \\ y &= \frac {1}{2} i x^{2}+c_{1} x +c_{2} \\ y &= \sinh \left (x +c_{1} \right )+c_{2} x +c_{3} \\ \end{align*}

✓ Mathematica. Time used: 0.251 (sec). Leaf size: 17

ode=D[y[x],{x,3}]== Sqrt[1+D[y[x],{x,2}]^2]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_3 x+\sinh (x+c_1)+c_2 \]

✓ Sympy. Time used: 0.346 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = C_{1} + C_{2} x + \sinh {\left (C_{3} + x \right )} \]