60.1.479 problem 492
Internal
problem
ID
[10493]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
492
Date
solved
:
Sunday, March 30, 2025 at 05:10:19 PM
CAS
classification
:
[_quadrature]
\begin{align*} \left (y^{2}-a^{2}\right ) {y^{\prime }}^{2}+y^{2}&=0 \end{align*}
✓ Maple. Time used: 0.079 (sec). Leaf size: 115
ode:=(y(x)^2-a^2)*diff(y(x),x)^2+y(x)^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
a \,\operatorname {csgn}\left (a \right ) \ln \left (\frac {a \left (\sqrt {-y^{2}+a^{2}}\, \operatorname {csgn}\left (a \right )+a \right )}{y}\right )+a \,\operatorname {csgn}\left (a \right ) \ln \left (2\right )-\sqrt {-y^{2}+a^{2}}-c_1 +x &= 0 \\
-a \,\operatorname {csgn}\left (a \right ) \ln \left (\frac {a \left (\sqrt {-y^{2}+a^{2}}\, \operatorname {csgn}\left (a \right )+a \right )}{y}\right )-a \,\operatorname {csgn}\left (a \right ) \ln \left (2\right )+\sqrt {-y^{2}+a^{2}}-c_1 +x &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.334 (sec). Leaf size: 102
ode=y[x]^2 + (-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 \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: 4.366 (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)
\[
\left [ \int \limits ^{y{\left (x \right )}} \frac {1}{y \sqrt {\frac {1}{- y^{2} + a^{2}}}}\, dy = C_{1} - x, \ \int \limits ^{y{\left (x \right )}} \frac {1}{y \sqrt {\frac {1}{- y^{2} + a^{2}}}}\, dy = C_{1} + x\right ]
\]