29.29.22 problem 844

Internal problem ID [5427]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 29
Problem number : 844
Date solved : Sunday, March 30, 2025 at 08:12:53 AM
CAS classification : [_quadrature]

\begin{align*} x {y^{\prime }}^{2}&=-x^{2}+a \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 47

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

\begin{align*} y &= \int \frac {\sqrt {x \left (-x^{2}+a \right )}}{x}d x +c_1 \\ y &= -\int \frac {\sqrt {x \left (-x^{2}+a \right )}}{x}d x +c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 113

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

\begin{align*} y(x)\to -\frac {2 \sqrt {x} \sqrt {a-x^2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {x^2}{a}\right )}{\sqrt {1-\frac {x^2}{a}}}+c_1 \\ y(x)\to \frac {2 \sqrt {x} \sqrt {a-x^2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {x^2}{a}\right )}{\sqrt {1-\frac {x^2}{a}}}+c_1 \\ \end{align*}

✓ Sympy. Time used: 0.604 (sec). Leaf size: 27

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

\[ \left [ y{\left (x \right )} = C_{1} - \int \sqrt {\frac {a}{x} - x}\, dx, \ y{\left (x \right )} = C_{1} + \int \sqrt {\frac {a}{x} - x}\, dx\right ] \]