23.1.728 problem 725

Internal problem ID [5335]
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 : 725
Date solved : Tuesday, September 30, 2025 at 12:30:42 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 67

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

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

✓ Mathematica. Time used: 0.597 (sec). Leaf size: 89

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\frac {1}{2} b^2 \text {arctanh}\left (\frac {\text {$\#$1}}{\sqrt {\text {$\#$1}^2+b^2}}\right )+\frac {1}{2} \text {$\#$1} \sqrt {\text {$\#$1}^2+b^2}\&\right ]\left [\frac {1}{2} \left (a^2 \text {arctanh}\left (\frac {x}{\sqrt {a^2+x^2}}\right )+x \sqrt {a^2+x^2}+2 c_1\right )\right ] \end{align*}

✓ Sympy. Time used: 3.423 (sec). Leaf size: 60

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

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