23.6.16 problem 16

Internal problem ID [6815]
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 : 16
Date solved : Tuesday, September 30, 2025 at 03:52:18 PM
CAS classification : [[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]]

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

✓ Maple. Time used: 0.305 (sec). Leaf size: 65

ode:=(1+diff(y(x),x)^2)*diff(diff(diff(y(x),x),x),x) = (a+3*diff(y(x),x))*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 &= \int \operatorname {RootOf}\left (-x \sqrt {\textit {\_Z}^{2}+1}\, {\mathrm e}^{a \arctan \left (\textit {\_Z} \right )}-c_2 \sqrt {\textit {\_Z}^{2}+1}\, {\mathrm e}^{a \arctan \left (\textit {\_Z} \right )}+c_1 \textit {\_Z} -c_1 a \right )d x +c_3 \\ \end{align*}

✓ Mathematica. Time used: 11.434 (sec). Leaf size: 198

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

\begin{align*} y(x)&\to c_3-\frac {\left (1-i \text {InverseFunction}\left [\frac {(\text {$\#$1}-a) e^{-a \arctan (\text {$\#$1})}}{\sqrt {\text {$\#$1}^2+1} \left (a^2+1\right ) c_1}\&\right ][x+c_2]\right ){}^{-\frac {1}{2}-\frac {i a}{2}} \left (1+i \text {InverseFunction}\left [\frac {(\text {$\#$1}-a) e^{-a \arctan (\text {$\#$1})}}{\sqrt {\text {$\#$1}^2+1} \left (a^2+1\right ) c_1}\&\right ][x+c_2]\right ){}^{\frac {1}{2} i (a+i)} \left (1+a \text {InverseFunction}\left [\frac {(\text {$\#$1}-a) e^{-a \arctan (\text {$\#$1})}}{\sqrt {\text {$\#$1}^2+1} \left (a^2+1\right ) c_1}\&\right ][x+c_2]\right )}{\left (a^2+1\right ) c_1}\\ y(x)&\to \text {Indeterminate}\\ y(x)&\to c_3 \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -(sqrt(4*a*Derivative(y(x), (x, 2))**2*Derivative(y(x), (x, 3))