29.31.9 problem 908
Internal
problem
ID
[5488]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
31
Problem
number
:
908
Date
solved
:
Tuesday, March 04, 2025 at 09:43:48 PM
CAS
classification
:
[_separable]
\begin{align*} x^{2} {y^{\prime }}^{2}-4 x \left (2+y\right ) y^{\prime }+4 \left (2+y\right ) y&=0 \end{align*}
✓ Maple. Time used: 0.056 (sec). Leaf size: 137
ode:=x^2*diff(y(x),x)^2-4*x*(2+y(x))*diff(y(x),x)+4*(2+y(x))*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= -2 \\
y \left (x \right ) &= \frac {-2 \sqrt {2}\, \sqrt {c_{1} x^{2}}+x^{2}}{c_{1}} \\
y \left (x \right ) &= \frac {2 \sqrt {2}\, \sqrt {c_{1} x^{2}}+x^{2}}{c_{1}} \\
y \left (x \right ) &= \frac {\left (-8 c_{1}^{2}+x^{2}\right ) \left (-2 \sqrt {2}\, c_{1} +x \right ) x}{\left (-4 c_{1} \sqrt {2}\, x +8 c_{1}^{2}+x^{2}\right ) c_{1}^{2}} \\
y \left (x \right ) &= \frac {\left (-8 c_{1}^{2}+x^{2}\right ) \left (2 \sqrt {2}\, c_{1} +x \right ) x}{\left (4 c_{1} \sqrt {2}\, x +8 c_{1}^{2}+x^{2}\right ) c_{1}^{2}} \\
\end{align*}
✓ Mathematica. Time used: 0.198 (sec). Leaf size: 69
ode=x^2 (D[y[x],x])^2-4 x(2+y[x])D[y[x],x]+4(2+y[x])y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to e^{-c_1} x \left (x-2 \sqrt {2} e^{\frac {c_1}{2}}\right ) \\
y(x)\to e^{c_1} x^2-2 \sqrt {2} e^{\frac {c_1}{2}} x \\
y(x)\to -2 \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 32.111 (sec). Leaf size: 160
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2*Derivative(y(x), x)**2 - 4*x*(y(x) + 2)*Derivative(y(x), x) + (4*y(x) + 8)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = x \left (- C_{1} x + 2 \sqrt {2} \sqrt {- C_{1}}\right ), \ y{\left (x \right )} = x \left (- C_{1} x - 2 \sqrt {2} \sqrt {- C_{1}}\right ), \ y{\left (x \right )} = x \left (- 2 \sqrt {2} \sqrt {C_{1}} + C_{1} x\right ), \ y{\left (x \right )} = x \left (2 \sqrt {2} \sqrt {C_{1}} + C_{1} x\right ), \ y{\left (x \right )} = x \left (- C_{1} x + 2 \sqrt {2} \sqrt {- C_{1}}\right ), \ y{\left (x \right )} = x \left (- C_{1} x - 2 \sqrt {2} \sqrt {- C_{1}}\right ), \ y{\left (x \right )} = x \left (- 2 \sqrt {2} \sqrt {C_{1}} + C_{1} x\right ), \ y{\left (x \right )} = x \left (2 \sqrt {2} \sqrt {C_{1}} + C_{1} x\right )\right ]
\]