54.1.361 problem 369

Internal problem ID [11675]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 369
Date solved : Tuesday, September 30, 2025 at 10:06:13 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 60

ode:=diff(y(x),x)^2+y(x)^2-a^2 = 0; 
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: 1.798 (sec). Leaf size: 62

ode=-a^2 + y[x]^2 + D[y[x],x]^2==0; 
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: 2.221 (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)
 

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