29.33.20 problem 982
Internal
problem
ID
[5559]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
33
Problem
number
:
982
Date
solved
:
Sunday, March 30, 2025 at 08:40:33 AM
CAS
classification
:
[_quadrature]
\begin{align*} \left (a^{2}-y^{2}\right ) {y^{\prime }}^{2}&=y^{2} \end{align*}
✓ Maple. Time used: 0.050 (sec). Leaf size: 115
ode:=(a^2-y(x)^2)*diff(y(x),x)^2 = y(x)^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
a \,\operatorname {csgn}\left (a \right ) \ln \left (\frac {a \left (\operatorname {csgn}\left (a \right ) \sqrt {a^{2}-y^{2}}+a \right )}{y}\right )+a \,\operatorname {csgn}\left (a \right ) \ln \left (2\right )-\sqrt {a^{2}-y^{2}}-c_1 +x &= 0 \\
-a \,\operatorname {csgn}\left (a \right ) \ln \left (\frac {a \left (\operatorname {csgn}\left (a \right ) \sqrt {a^{2}-y^{2}}+a \right )}{y}\right )-a \,\operatorname {csgn}\left (a \right ) \ln \left (2\right )+\sqrt {a^{2}-y^{2}}-c_1 +x &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.336 (sec). Leaf size: 102
ode=(a^2-y[x]^2) (D[y[x],x])^2==y[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\sqrt {a^2-\text {$\#$1}^2}-a \text {arctanh}\left (\frac {\sqrt {a^2-\text {$\#$1}^2}}{a}\right )\&\right ][-x+c_1] \\
y(x)\to \text {InverseFunction}\left [\sqrt {a^2-\text {$\#$1}^2}-a \text {arctanh}\left (\frac {\sqrt {a^2-\text {$\#$1}^2}}{a}\right )\&\right ][x+c_1] \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 4.924 (sec). Leaf size: 41
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq((a**2 - y(x)**2)*Derivative(y(x), x)**2 - y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \int \limits ^{y{\left (x \right )}} \frac {1}{y \sqrt {\frac {1}{- y^{2} + a^{2}}}}\, dy = C_{1} - x, \ \int \limits ^{y{\left (x \right )}} \frac {1}{y \sqrt {\frac {1}{- y^{2} + a^{2}}}}\, dy = C_{1} + x\right ]
\]