23.1.411 problem 398

Internal problem ID [5018]
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 : 398
Date solved : Tuesday, September 30, 2025 at 11:23:25 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 79

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

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

✓ Mathematica. Time used: 27.306 (sec). Leaf size: 276

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

\begin{align*} y(x)&\to -\frac {2 i b^{3/2} e^{b c_1} \left (a \left (a-\sqrt {a^2+x^2}\right )\right )^{\frac {b}{2 a}} \left (\sqrt {a^2+x^2}+a\right )^{\frac {b}{2 a}}}{\sqrt {\left (b \left (\sqrt {a^2+x^2}+a\right )^{\frac {b}{a}}+e^{2 b c_1} \left (a \left (a-\sqrt {a^2+x^2}\right )\right )^{\frac {b}{a}}\right ){}^2}}\\ y(x)&\to \frac {2 i b^{3/2} e^{b c_1} \left (a \left (a-\sqrt {a^2+x^2}\right )\right )^{\frac {b}{2 a}} \left (\sqrt {a^2+x^2}+a\right )^{\frac {b}{2 a}}}{\sqrt {\left (b \left (\sqrt {a^2+x^2}+a\right )^{\frac {b}{a}}+e^{2 b c_1} \left (a \left (a-\sqrt {a^2+x^2}\right )\right )^{\frac {b}{a}}\right ){}^2}}\\ y(x)&\to 0\\ y(x)&\to -i b\\ y(x)&\to i b \end{align*}

✓ Sympy. Time used: 0.778 (sec). Leaf size: 17

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

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