29.30.35 problem 895
Internal
problem
ID
[5475]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
30
Problem
number
:
895
Date
solved
:
Sunday, March 30, 2025 at 08:15:26 AM
CAS
classification
:
[_separable]
\begin{align*} x^{2} {y^{\prime }}^{2}+y^{2}-y^{4}&=0 \end{align*}
✓ Maple. Time used: 0.051 (sec). Leaf size: 52
ode:=x^2*diff(y(x),x)^2+y(x)^2-y(x)^4 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -1 \\
y &= 1 \\
y &= 0 \\
y &= \operatorname {csgn}\left (\sec \left (-\ln \left (x \right )+c_1 \right )\right ) \csc \left (-\ln \left (x \right )+c_1 \right ) \\
y &= -\operatorname {csgn}\left (\sec \left (-\ln \left (x \right )+c_1 \right )\right ) \csc \left (-\ln \left (x \right )+c_1 \right ) \\
\end{align*}
✓ Mathematica. Time used: 1.478 (sec). Leaf size: 88
ode=x^2 (D[y[x],x])^2+y[x]^2-y[x]^4==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\sqrt {\sec ^2(-\log (x)+c_1)} \\
y(x)\to \sqrt {\sec ^2(-\log (x)+c_1)} \\
y(x)\to -\sqrt {\sec ^2(\log (x)+c_1)} \\
y(x)\to \sqrt {\sec ^2(\log (x)+c_1)} \\
y(x)\to -1 \\
y(x)\to 0 \\
y(x)\to 1 \\
\end{align*}
✓ Sympy. Time used: 0.879 (sec). Leaf size: 54
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2*Derivative(y(x), x)**2 - y(x)**4 + y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \begin {cases} i \operatorname {acosh}{\left (\frac {1}{y{\left (x \right )}} \right )} & \text {for}\: \frac {1}{\left |{y^{2}{\left (x \right )}}\right |} > 1 \\- \operatorname {asin}{\left (\frac {1}{y{\left (x \right )}} \right )} & \text {otherwise} \end {cases} = C_{1} - \log {\left (x \right )}, \ \begin {cases} i \operatorname {acosh}{\left (\frac {1}{y{\left (x \right )}} \right )} & \text {for}\: \frac {1}{\left |{y^{2}{\left (x \right )}}\right |} > 1 \\- \operatorname {asin}{\left (\frac {1}{y{\left (x \right )}} \right )} & \text {otherwise} \end {cases} = C_{1} + \log {\left (x \right )}\right ]
\]