83.35.6 problem 6

Internal problem ID [19347]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Misc. Exercise on chapter VII. Page 118
Problem number : 6
Date solved : Thursday, March 13, 2025 at 02:16:37 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.052 (sec). Leaf size: 79

ode:=diff(diff(y(x),x),x) = 1/(a*y(x))^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \int _{}^{y \left (x \right )}\frac {\sqrt {\textit {\_a} a}}{\sqrt {\sqrt {\textit {\_a} a}\, \sqrt {\textit {\_a}}\, \left (c_{1} +4 \sqrt {\textit {\_a}}\right )}}d \textit {\_a} -x -c_{2} &= 0 \\ -\int _{}^{y \left (x \right )}\frac {\sqrt {\textit {\_a} a}}{\sqrt {\sqrt {\textit {\_a} a}\, \sqrt {\textit {\_a}}\, \left (c_{1} +4 \sqrt {\textit {\_a}}\right )}}d \textit {\_a} -x -c_{2} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 60.067 (sec). Leaf size: 1570

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

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

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

TypeError : < not supported between instances of NoneType and y