82.18.15 problem Ex. 16

Internal problem ID [18766]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter III. Equations of the first order but not of the first degree. Examples on chapter III. page 38
Problem number : Ex. 16
Date solved : Monday, March 31, 2025 at 06:09:45 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} \left ({y^{\prime }}^{2}-\frac {1}{a^{2}-x^{2}}\right ) \left (y^{\prime }-\sqrt {\frac {y}{x}}\right )&=0 \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 75

ode:=(diff(y(x),x)^2-1/(a^2-x^2))*(diff(y(x),x)-(y(x)/x)^(1/2)) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} -\ln \left (\frac {y-x}{x}\right )+2 \,\operatorname {arctanh}\left (\sqrt {\frac {y}{x}}\right )-\ln \left (x \right )-c_1 &= 0 \\ y &= \arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )+c_1 \\ y &= -\arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.078 (sec). Leaf size: 69

ode=(D[y[x],x]^2- 1/(a^2-x^2))*( D[y[x],x] - Sqrt[ y[x]/x] )==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 3.091 (sec). Leaf size: 53

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

\[ \left [ y{\left (x \right )} = \frac {C_{1}^{2}}{4} - C_{1} \sqrt {x} + x, \ y{\left (x \right )} = C_{1} - \int \sqrt {- \frac {1}{- a^{2} + x^{2}}}\, dx, \ y{\left (x \right )} = C_{1} + \int \sqrt {- \frac {1}{- a^{2} + x^{2}}}\, dx\right ] \]