76.15.4 problem Ex. 4
Internal
problem
ID
[20099]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
III.
Equations
of
the
first
order
but
not
of
the
first
degree.
Problems
at
page
35
Problem
number
:
Ex.
4
Date
solved
:
Thursday, October 02, 2025 at 05:25:44 PM
CAS
classification
:
[_quadrature]
\begin{align*} y^{2}&=a^{2} \left (1+{y^{\prime }}^{2}\right ) \end{align*}
✓ Maple. Time used: 0.052 (sec). Leaf size: 73
ode:=y(x)^2 = a^2*(1+diff(y(x),x)^2);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -a \\
y &= a \\
y &= \frac {{\mathrm e}^{\frac {c_1 -x}{a}} a^{2}}{2}+\frac {{\mathrm e}^{\frac {-c_1 +x}{a}}}{2} \\
y &= \frac {{\mathrm e}^{\frac {c_1 -x}{a}}}{2}+\frac {{\mathrm e}^{\frac {-c_1 +x}{a}} a^{2}}{2} \\
\end{align*}
✓ Mathematica. Time used: 23.98 (sec). Leaf size: 82
ode=y[x]^2==a^2*(1+D[y[x],x]^2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {a \tanh \left (\frac {x}{a}-c_1\right )}{\sqrt {-\text {sech}^2\left (\frac {x}{a}-c_1\right )}}\\ y(x)&\to -\frac {a \tanh \left (\frac {x}{a}+c_1\right )}{\sqrt {-\text {sech}^2\left (\frac {x}{a}+c_1\right )}}\\ y(x)&\to -a\\ y(x)&\to a \end{align*}
✓ Sympy. Time used: 2.673 (sec). Leaf size: 97
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a**2*(Derivative(y(x), x)**2 + 1) + y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \begin {cases} a^{2} e^{- C_{1} + \frac {x}{a}} + \frac {e^{C_{1} - \frac {x}{a}}}{4} & \text {for}\: a^{2} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} e^{\frac {x}{a}} & \text {for}\: a^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} e^{- \frac {x}{a}} & \text {for}\: a^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} a^{2} e^{- C_{1} - \frac {x}{a}} + \frac {e^{C_{1} + \frac {x}{a}}}{4} & \text {for}\: a^{2} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} e^{- \frac {x}{a}} & \text {for}\: a^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} e^{\frac {x}{a}} & \text {for}\: a^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}\right ]
\]