82.15.3 problem Ex. 3

Internal problem ID [18742]
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. Problems at page 35
Problem number : Ex. 3
Date solved : Monday, March 31, 2025 at 06:07:21 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 91

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

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

✓ Mathematica. Time used: 0.044 (sec). Leaf size: 99

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

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

✓ Sympy. Time used: 1.329 (sec). Leaf size: 105

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

\[ \left [ y{\left (x \right )} = C_{1} - \frac {- a^{2} \left (\begin {cases} \log {\left (2 x + 2 \sqrt {- a^{2} + x^{2}} \right )} & \text {for}\: a^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {x^{2}}} & \text {otherwise} \end {cases}\right ) + x \sqrt {- a^{2} + x^{2}}}{2 a}, \ y{\left (x \right )} = C_{1} + \frac {- a^{2} \left (\begin {cases} \log {\left (2 x + 2 \sqrt {- a^{2} + x^{2}} \right )} & \text {for}\: a^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {x^{2}}} & \text {otherwise} \end {cases}\right ) + x \sqrt {- a^{2} + x^{2}}}{2 a}\right ] \]