29.31.17 problem 916

Internal problem ID [5496]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 31
Problem number : 916
Date solved : Sunday, March 30, 2025 at 08:25:21 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 40

ode:=(a^2+x^2)*diff(y(x),x)^2 = b^2; 
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: 48

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

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

✓ Sympy. Time used: 0.674 (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)
 

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