23.1.409 problem 396

Internal problem ID [5016]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 396
Date solved : Tuesday, September 30, 2025 at 11:23:06 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 33

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

\[ \ln \left (x +\sqrt {b^{2}+x^{2}}\right )-\ln \left (y+\sqrt {y^{2}+a^{2}}\right )+c_1 = 0 \]

✓ Mathematica. Time used: 29.697 (sec). Leaf size: 68

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

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

✓ Sympy. Time used: 0.604 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = a \sinh {\left (C_{1} + \operatorname {asinh}{\left (\frac {x}{b} \right )} \right )} \]