29.14.12 problem 393

Internal problem ID [4991]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 14
Problem number : 393
Date solved : Tuesday, March 04, 2025 at 07:40:36 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.002 (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);
 

\[ \ln \left (x +\sqrt {b^{2}+x^{2}}\right )-\ln \left (y \left (x \right )+\sqrt {a^{2}+y \left (x \right )^{2}}\right )+c_{1} = 0 \]

✓ Mathematica. Time used: 0.384 (sec). Leaf size: 76

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^2 e^{-c_1} \left (x-\sqrt {b^2+x^2}\right )}{2 b^2}+\frac {1}{2} e^{c_1} \left (\sqrt {b^2+x^2}+x\right ) \\ y(x)\to -i a \\ y(x)\to i a \\ \end{align*}

✓ Sympy. Time used: 0.958 (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)
 

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