29.26.21 problem 757
Internal
problem
ID
[5342]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
26
Problem
number
:
757
Date
solved
:
Sunday, March 30, 2025 at 08:00:44 AM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{2}&=a^{2}-y^{2} \end{align*}
✓ Maple. Time used: 0.033 (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: 3.514 (sec). Leaf size: 111
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 -\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: 3.589 (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)
\[
\left [ y{\left (x \right )} = \begin {cases} - a^{2} e^{- i C_{1} + i x} - \frac {e^{i C_{1} - i x}}{4} & \text {for}\: a^{2} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} e^{i x} & \text {for}\: a^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} e^{- i x} & \text {for}\: a^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} - a^{2} e^{- i C_{1} - i x} - \frac {e^{i C_{1} + i x}}{4} & \text {for}\: a^{2} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} e^{- i x} & \text {for}\: a^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} e^{i x} & \text {for}\: a^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}\right ]
\]