23.4.75 problem 75

Internal problem ID [6377]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 75
Date solved : Tuesday, September 30, 2025 at 02:54:51 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 9 {y^{\prime }}^{4}+8 y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 57

ode:=9*diff(y(x),x)^4+8*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.153 (sec). Leaf size: 90

ode=9*D[y[x],x]^4 + 8*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2-\frac {1}{3} \sqrt [3]{-\frac {1}{3}} (9 x-8 c_1){}^{2/3}\\ y(x)&\to \frac {(9 x-8 c_1){}^{2/3}}{3 \sqrt [3]{3}}+c_2\\ y(x)&\to \frac {1}{9} \left ((-3)^{2/3} (9 x-8 c_1){}^{2/3}+9 c_2\right ) \end{align*}

✓ Sympy. Time used: 13.426 (sec). Leaf size: 100

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*Derivative(y(x), x)**4 + 8*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = C_{1} + \frac {\left (- \frac {3^{\frac {2}{3}}}{3} - \sqrt [6]{3} i\right ) \left (C_{2} + 9 x\right ) \sqrt [3]{\frac {1}{C_{2} + 9 x}}}{6}, \ y{\left (x \right )} = C_{1} + \frac {\left (- \frac {3^{\frac {2}{3}}}{3} + \sqrt [6]{3} i\right ) \left (C_{2} + 9 x\right ) \sqrt [3]{\frac {1}{C_{2} + 9 x}}}{6}, \ y{\left (x \right )} = C_{1} + \frac {C_{2} \sqrt [3]{\frac {1}{C_{2} + 27 x}}}{9} + 3 x \sqrt [3]{\frac {1}{C_{2} + 27 x}}\right ] \]