82.15.4 problem Ex. 4
Internal
problem
ID
[18735]
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, March 13, 2025 at 12:44:45 PM
CAS
classification
:
[_quadrature]
\begin{align*} y^{2}&=a^{2} \left (1+{y^{\prime }}^{2}\right ) \end{align*}
✓ Maple. Time used: 0.021 (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 \left (x \right ) &= -a \\
y \left (x \right ) &= a \\
y \left (x \right ) &= \frac {a^{2} {\mathrm e}^{\frac {-x +c_{1}}{a}}}{2}+\frac {{\mathrm e}^{\frac {-c_{1} +x}{a}}}{2} \\
y \left (x \right ) &= \frac {{\mathrm e}^{\frac {-x +c_{1}}{a}}}{2}+\frac {a^{2} {\mathrm e}^{\frac {-c_{1} +x}{a}}}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.353 (sec). Leaf size: 86
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 {1}{2} \left (a^2 e^{\frac {x}{a}-c_1}+e^{-\frac {x}{a}+c_1}\right ) \\
y(x)\to \frac {1}{2} e^{-\frac {x+a c_1}{a}} \left (a^2+e^{2 \left (\frac {x}{a}+c_1\right )}\right ) \\
y(x)\to -a \\
y(x)\to a \\
\end{align*}
✓ Sympy. Time used: 4.226 (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)
\[
\left [ y{\left (x \right )} = \begin {cases} a^{2} e^{- C_{1} + \frac {x}{a}} + \frac {e^{C_{1} - \frac {x}{a}}}{4} & \text {for}\: a^{2} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} e^{\frac {x}{a}} & \text {for}\: a^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} e^{- \frac {x}{a}} & \text {for}\: a^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} a^{2} e^{- C_{1} - \frac {x}{a}} + \frac {e^{C_{1} + \frac {x}{a}}}{4} & \text {for}\: a^{2} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} e^{- \frac {x}{a}} & \text {for}\: a^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} e^{\frac {x}{a}} & \text {for}\: a^{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}\right ]
\]