50.5.1 problem 1

Internal problem ID [10248]
Book : Own collection of miscellaneous problems
Section : section 5.0
Problem number : 1
Date solved : Tuesday, September 30, 2025 at 07:13:14 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y^{\prime \prime }&=A y^{{2}/{3}} \end{align*}

✓ Maple. Time used: 0.047 (sec). Leaf size: 61

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

\begin{align*} y &= 0 \\ -5 \int _{}^{y}\frac {1}{\sqrt {30 \textit {\_a}^{{5}/{3}} A -5 c_1}}d \textit {\_a} -x -c_2 &= 0 \\ 5 \int _{}^{y}\frac {1}{\sqrt {30 \textit {\_a}^{{5}/{3}} A -5 c_1}}d \textit {\_a} -x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.056 (sec). Leaf size: 75

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

\[ \text {Solve}\left [\frac {y(x)^2 \left (1+\frac {6 A y(x)^{5/3}}{5 c_1}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{5},\frac {8}{5},-\frac {6 A y(x)^{5/3}}{5 c_1}\right ){}^2}{\frac {6}{5} A y(x)^{5/3}+c_1}=(x+c_2){}^2,y(x)\right ] \]

✓ Sympy. Time used: 8.429 (sec). Leaf size: 99

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

\[ \left [ \frac {3 y{\left (x \right )} \Gamma \left (\frac {3}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{5} \\ \frac {8}{5} \end {matrix}\middle | {\frac {6 A e^{i \pi } y^{\frac {5}{3}}{\left (x \right )}}{5 C_{1}}} \right )}}{5 \sqrt {C_{1}} \Gamma \left (\frac {8}{5}\right )} = C_{2} + x, \ \frac {3 y{\left (x \right )} \Gamma \left (\frac {3}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{5} \\ \frac {8}{5} \end {matrix}\middle | {\frac {6 A e^{i \pi } y^{\frac {5}{3}}{\left (x \right )}}{5 C_{1}}} \right )}}{5 \sqrt {C_{1}} \Gamma \left (\frac {8}{5}\right )} = C_{2} - x\right ] \]