48.3.2 problem Example 3.30

Internal problem ID [7526]
Book : THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section : Chapter 3. Ordinary Differential Equations. Section 3.5 HIGHER ORDER ODE. Page 181
Problem number : Example 3.30
Date solved : Sunday, March 30, 2025 at 12:13:40 PM
CAS classification : [[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_exponential_symmetries], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]]

\begin{align*} 3 {y^{\prime \prime }}^{2}-y^{\prime } y^{\prime \prime \prime }-y^{\prime \prime } {y^{\prime }}^{2}&=0 \end{align*}

✓ Maple. Time used: 0.055 (sec). Leaf size: 38

ode:=3*diff(diff(y(x),x),x)^2-diff(y(x),x)*diff(diff(diff(y(x),x),x),x)-diff(diff(y(x),x),x)*diff(y(x),x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= c_1 \\ y &= \frac {\operatorname {LambertW}\left (-\frac {{\mathrm e}^{\frac {c_3 +x}{c_1}}}{c_2 c_1}\right ) c_1 -c_3 -x}{c_1} \\ \end{align*}

✓ Mathematica. Time used: 4.035 (sec). Leaf size: 79

ode=3*(D[y[x],{x,2}])^2-D[y[x],x]*D[y[x],{x,3}]-D[y[x],{x,2}]*(D[y[x],x])^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \log \left (\text {InverseFunction}\left [-\frac {1}{\text {$\#$1}}-c_1 \log (\text {$\#$1})+c_1 \log (1+\text {$\#$1} c_1)\&\right ][x+c_2]\right )-\log \left (1+c_1 \text {InverseFunction}\left [-\frac {1}{\text {$\#$1}}-c_1 \log (\text {$\#$1})+c_1 \log (1+\text {$\#$1} c_1)\&\right ][x+c_2]\right )+c_3 \]

✓ Sympy. Time used: 2.076 (sec). Leaf size: 7

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

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