23.6.17 problem 17

Internal problem ID [6816]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 6. THE EQUATION IS NONLINEAR AND OF ORDER GREATER THAN TWO, page 427
Problem number : 17
Date solved : Tuesday, September 30, 2025 at 03:52:22 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 }}^{3} y^{\prime \prime \prime }&=1 \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 151

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

\begin{align*} y &= \frac {\ln \left (\frac {c_1^{3} \left (c_2 +x \right )}{\sqrt {c_1^{3}}}+\sqrt {\left (1+\left (c_2 +x \right )^{2} c_1^{2}\right ) c_1}\right ) c_1 +\sqrt {c_1^{3}}\, \left (\left (c_2 +x \right ) \sqrt {\left (1+\left (c_2 +x \right )^{2} c_1^{2}\right ) c_1}+2 c_1 c_3 \right )}{2 \sqrt {c_1^{3}}\, c_1} \\ y &= -\frac {\ln \left (\frac {c_1^{3} \left (c_2 +x \right )}{\sqrt {c_1^{3}}}+\sqrt {\left (1+\left (c_2 +x \right )^{2} c_1^{2}\right ) c_1}\right ) c_1 +\sqrt {c_1^{3}}\, \left (\left (c_2 +x \right ) \sqrt {\left (1+\left (c_2 +x \right )^{2} c_1^{2}\right ) c_1}-2 c_1 c_3 \right )}{2 \sqrt {c_1^{3}}\, c_1} \\ \end{align*}

✓ Mathematica. Time used: 14.329 (sec). Leaf size: 136

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

\begin{align*} y(x)&\to c_3-\frac {\text {arcsinh}(c_1 (x+c_2))+c_1 (x+c_2) \sqrt {c_1{}^2 x^2+2 c_2 c_1{}^2 x+1+c_2{}^2 c_1{}^2}}{2 c_1{}^{3/2}}\\ y(x)&\to \frac {\text {arcsinh}(c_1 (x+c_2))+c_1 (x+c_2) \sqrt {c_1{}^2 x^2+2 c_2 c_1{}^2 x+1+c_2{}^2 c_1{}^2}}{2 c_1{}^{3/2}}+c_3\\ y(x)&\to \text {Indeterminate} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -(1/Derivative(y(x), (x, 3)))**(1/3) + Derivative(y(x), x) canno