56.5.1 problem 1

Internal problem ID [8962]
Book : Own collection of miscellaneous problems
Section : section 5.0
Problem number : 1
Date solved : Monday, January 27, 2025 at 05:22:49 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Solution by Maple

Time used: 0.055 (sec). Leaf size: 61

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

\begin{align*} y &= 0 \\ -5 \left (\int _{}^{y}\frac {1}{\sqrt {30 \textit {\_a}^{{5}/{3}} A -5 c_{1}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ 5 \left (\int _{}^{y}\frac {1}{\sqrt {30 \textit {\_a}^{{5}/{3}} A -5 c_{1}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.097 (sec). Leaf size: 75

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

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