38.1.45 problem 47

Internal problem ID [8206]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 1. Introduction to differential equations. Exercises 1.1 at page 12
Problem number : 47
Date solved : Tuesday, September 30, 2025 at 05:18:47 PM
CAS classification : [_quadrature]

\begin{align*} y y^{\prime }+\sqrt {16-y^{2}}&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 25

ode:=y(x)*diff(y(x),x)+(16-y(x)^2)^(1/2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ x +\frac {\left (y-4\right ) \left (y+4\right )}{\sqrt {16-y^{2}}}+c_1 = 0 \]

✓ Mathematica. Time used: 0.17 (sec). Leaf size: 65

ode=y[x]*D[y[x],x]+Sqrt[16-y[x]^2]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\sqrt {-x^2+2 c_1 x+16-c_1{}^2}\\ y(x)&\to \sqrt {-x^2+2 c_1 x+16-c_1{}^2}\\ y(x)&\to -4\\ y(x)&\to 4 \end{align*}

✓ Sympy. Time used: 0.678 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(sqrt(16 - y(x)**2) + y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \sqrt {- C_{1}^{2} + 2 C_{1} x - x^{2} + 16}, \ y{\left (x \right )} = \sqrt {- C_{1}^{2} + 2 C_{1} x - x^{2} + 16}\right ] \]