23.1.442 problem 432

Internal problem ID [5049]
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 : 432
Date solved : Tuesday, September 30, 2025 at 11:30:14 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 18

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

\[ x -\sqrt {y^{2}+a^{2}}+c_1 = 0 \]

✓ Mathematica. Time used: 0.145 (sec). Leaf size: 61

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

\begin{align*} y(x)&\to -\sqrt {-a^2+(x+c_1){}^2}\\ y(x)&\to \sqrt {-a^2+(x+c_1){}^2}\\ y(x)&\to -i a\\ y(x)&\to i a \end{align*}

✓ Sympy. Time used: 0.625 (sec). Leaf size: 42

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

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