42.4.11 problem Problem 3.18

Internal problem ID [8839]
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 3.6 Summary and Problems. Page 218
Problem number : Problem 3.18
Date solved : Tuesday, September 30, 2025 at 05:57:29 PM
CAS classification : [[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]

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

✓ Maple. Time used: 0.083 (sec). Leaf size: 173

ode:=a*diff(diff(y(x),x),x)*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 &= \frac {\left (2 a^{2}+\left (c_1 +x \right )^{2}\right ) \sqrt {c_1^{2}+2 c_1 x -a^{2}+x^{2}}-3 a \left (a \left (c_1 +x \right ) \ln \left (c_1 +x +\sqrt {\left (c_1 +a +x \right ) \left (c_1 -a +x \right )}\right )-2 c_2 x -2 c_3 \right )}{6 a} \\ y &= \frac {\left (-2 a^{2}-\left (c_1 +x \right )^{2}\right ) \sqrt {c_1^{2}+2 c_1 x -a^{2}+x^{2}}+3 a \left (a \left (c_1 +x \right ) \ln \left (c_1 +x +\sqrt {\left (c_1 +a +x \right ) \left (c_1 -a +x \right )}\right )+2 c_2 x +2 c_3 \right )}{6 a} \\ \end{align*}

✓ Mathematica. Time used: 60.319 (sec). Leaf size: 197

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

\begin{align*} y(x)&\to \int _1^x\left (\frac {1}{2} a \log \left (a c_1+K[1]+\sqrt {\left (c_1{}^2-1\right ) a^2+2 c_1 K[1] a+K[1]^2}\right )-\frac {(a c_1+K[1]) \sqrt {\left (c_1{}^2-1\right ) a^2+2 c_1 K[1] a+K[1]^2}}{2 a}\right )dK[1]+c_3 x+c_2\\ y(x)&\to \int _1^x\left (\frac {(a c_1+K[2]) \sqrt {\left (c_1{}^2-1\right ) a^2+2 c_1 K[2] a+K[2]^2}}{2 a}-\frac {1}{2} a \log \left (a c_1+K[2]+\sqrt {\left (c_1{}^2-1\right ) a^2+2 c_1 K[2] a+K[2]^2}\right )\right )dK[2]+c_3 x+c_2 \end{align*}

✗ Sympy

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

Timed Out