67.8.18 problem 13.3 (c)

Internal problem ID [16513]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 13. Higher order equations: Extending first order concepts. Additional exercises page 259
Problem number : 13.3 (c)
Date solved : Thursday, October 02, 2025 at 01:35:51 PM
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 }&=2 \sqrt {y^{\prime \prime }} \end{align*}

✓ Maple. Time used: 0.733 (sec). Leaf size: 36

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

\begin{align*} y &= c_{1} x +c_{2} \\ y &= \frac {1}{12} x^{4}+\frac {1}{3} c_{1} x^{3}+\frac {1}{2} c_{1}^{2} x^{2}+c_{2} x +c_{3} \\ \end{align*}

✓ Mathematica. Time used: 0.079 (sec). Leaf size: 39

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

\begin{align*} y(x)&\to \frac {x^4}{12}+\frac {c_1 x^3}{6}+\frac {c_1{}^2 x^2}{8}+c_3 x+c_2 \end{align*}

✓ Sympy. Time used: 0.183 (sec). Leaf size: 27

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

\[ y{\left (x \right )} = C_{1} + C_{2} x + \frac {C_{3}^{2} x^{2}}{8} + \frac {C_{3} x^{3}}{6} + \frac {x^{4}}{12} \]