29.30.33 problem 893
Internal
problem
ID
[5473]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
30
Problem
number
:
893
Date
solved
:
Tuesday, March 04, 2025 at 09:40:41 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} x^{2} {y^{\prime }}^{2}+x^{2}-y^{2}&=0 \end{align*}
✓ Maple. Time used: 0.055 (sec). Leaf size: 44
ode:=x^2*diff(y(x),x)^2+x^2-y(x)^2 = 0;
dsolve(ode,y(x), singsol=all);
\[
y \left (x \right ) = \frac {x \left (\operatorname {LambertW}\left (-{\mathrm e} c_{1} x^{4}\right )-1\right )}{2 \operatorname {LambertW}\left (-{\mathrm e} c_{1} x^{4}\right ) \sqrt {-\frac {1}{\operatorname {LambertW}\left (-{\mathrm e} c_{1} x^{4}\right )}}}
\]
✓ Mathematica. Time used: 7.539 (sec). Leaf size: 183
ode=x^2 (D[y[x],x])^2+x^2-y[x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [\text {arctanh}\left (\frac {\sqrt {\frac {y(x)}{x}+1}}{\sqrt {\frac {y(x)}{x}-1}}\right )-\frac {\sqrt {\frac {y(x)}{x}-1} \sqrt {\frac {y(x)}{x}+1}}{\left (\sqrt {\frac {y(x)}{x}-1}-\sqrt {\frac {y(x)}{x}+1}\right )^2}&=\log (x)+c_1,y(x)\right ] \\
\text {Solve}\left [\text {arctanh}\left (\frac {\sqrt {\frac {y(x)}{x}+1}}{\sqrt {\frac {y(x)}{x}-1}}\right )-\frac {\sqrt {\frac {y(x)}{x}-1} \sqrt {\frac {y(x)}{x}+1}}{\left (\sqrt {\frac {y(x)}{x}-1}+\sqrt {\frac {y(x)}{x}+1}\right )^2}&=-\log (x)+c_1,y(x)\right ] \\
\end{align*}
✓ Sympy. Time used: 14.360 (sec). Leaf size: 61
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2*Derivative(y(x), x)**2 + x**2 - y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} e^{- \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\sqrt {1 - u_{1}^{2}}}{u_{1} \left (\sqrt {1 - u_{1}^{2}} + 1\right )}\, du_{1}}, \ y{\left (x \right )} = C_{1} e^{- \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\sqrt {1 - u_{1}^{2}}}{u_{1} \left (\sqrt {1 - u_{1}^{2}} - 1\right )}\, du_{1}}\right ]
\]