42.4.2 problem Problem 3.2

Internal problem ID [8830]
Book : THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section : Chapter 3. Ordinary Differential Equations. Section 3.6 Summary and Problems. Page 218
Problem number : Problem 3.2
Date solved : Tuesday, September 30, 2025 at 05:53:04 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 60

ode:=diff(y(x),x)^2 = a^2-y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -a \\ y &= a \\ y &= -\tan \left (-x +c_1 \right ) \sqrt {a^{2} \cos \left (-x +c_1 \right )^{2}} \\ y &= \tan \left (-x +c_1 \right ) \sqrt {a^{2} \cos \left (-x +c_1 \right )^{2}} \\ \end{align*}

✓ Mathematica. Time used: 1.766 (sec). Leaf size: 62

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

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

✓ Sympy. Time used: 2.201 (sec). Leaf size: 107

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

\[ \left [ y{\left (x \right )} = \begin {cases} - a^{2} e^{- i C_{1} + i x} - \frac {e^{i C_{1} - i x}}{4} & \text {for}\: a^{2} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} e^{i x} & \text {for}\: a^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} e^{- i x} & \text {for}\: a^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} - a^{2} e^{- i C_{1} - i x} - \frac {e^{i C_{1} + i x}}{4} & \text {for}\: a^{2} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} e^{- i x} & \text {for}\: a^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} e^{i x} & \text {for}\: a^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}\right ] \]