29.31.18 problem 917

Internal problem ID [5497]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 31
Problem number : 917
Date solved : Sunday, March 30, 2025 at 08:25:22 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.039 (sec). Leaf size: 44

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

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 52

ode=(a^2-x^2) (D[y[x],x])^2+b^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -b \log \left (\sqrt {x^2-a^2}+x\right )+c_1 \\ y(x)\to b \log \left (\sqrt {x^2-a^2}+x\right )+c_1 \\ \end{align*}

✓ Sympy. Time used: 1.275 (sec). Leaf size: 37

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

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