23.2.239 problem 245
Internal
problem
ID
[5594]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
2.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
SECOND
OR
HIGHER
DEGREE,
page
278
Problem
number
:
245
Date
solved
:
Tuesday, September 30, 2025 at 01:10:46 PM
CAS
classification
:
[[_homogeneous, `class A`], _dAlembert]
\begin{align*} 3 y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }-x^{2}+4 y^{2}&=0 \end{align*}
✓ Maple. Time used: 0.082 (sec). Leaf size: 105
ode:=3*y(x)^2*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)-x^2+4*y(x)^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {\sqrt {3}\, x}{3} \\
y &= \frac {\sqrt {3}\, x}{3} \\
\ln \left (x \right )-\operatorname {arctanh}\left (\frac {\sqrt {\frac {x^{2}-3 y^{2}}{x^{2}}}}{2}\right )+\frac {\ln \left (\frac {x^{2}+y^{2}}{x^{2}}\right )}{2}-c_1 &= 0 \\
\ln \left (x \right )+\operatorname {arctanh}\left (\frac {\sqrt {\frac {x^{2}-3 y^{2}}{x^{2}}}}{2}\right )+\frac {\ln \left (\frac {x^{2}+y^{2}}{x^{2}}\right )}{2}-c_1 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.33 (sec). Leaf size: 179
ode=3 y[x]^2 (D[y[x],x])^2 -2 x y[x] D[y[x],x]-x^2+4 y[x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {\sqrt {-3 x^2-4 i e^{3 c_1} x+e^{6 c_1}}}{\sqrt {3}}\\ y(x)&\to \frac {\sqrt {-3 x^2-4 i e^{3 c_1} x+e^{6 c_1}}}{\sqrt {3}}\\ y(x)&\to -\frac {\sqrt {-3 x^2+4 i e^{3 c_1} x+e^{6 c_1}}}{\sqrt {3}}\\ y(x)&\to \frac {\sqrt {-3 x^2+4 i e^{3 c_1} x+e^{6 c_1}}}{\sqrt {3}}\\ y(x)&\to -\sqrt {-x^2}\\ y(x)&\to \sqrt {-x^2} \end{align*}
✓ Sympy. Time used: 148.974 (sec). Leaf size: 551
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**2 - 2*x*y(x)*Derivative(y(x), x) + 3*y(x)**2*Derivative(y(x), x)**2 + 4*y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {6} \sqrt {- 4 \sqrt {2} \sqrt {C_{1}} x - C_{1} - 6 x^{2}}}{6}, \ y{\left (x \right )} = \frac {\sqrt {6} \sqrt {- 4 \sqrt {2} \sqrt {C_{1}} x - C_{1} - 6 x^{2}}}{6}, \ y{\left (x \right )} = - \frac {\sqrt {6} \sqrt {C_{1} - 6 x^{2} + 4 \sqrt {2} x \sqrt {- C_{1}}}}{6}, \ y{\left (x \right )} = \frac {\sqrt {6} \sqrt {C_{1} - 6 x^{2} + 4 \sqrt {2} x \sqrt {- C_{1}}}}{6}, \ y{\left (x \right )} = - \frac {\sqrt {6} \sqrt {- 4 \sqrt {2} \sqrt {C_{1}} x - C_{1} - 6 x^{2}}}{6}, \ y{\left (x \right )} = \frac {\sqrt {6} \sqrt {- 4 \sqrt {2} \sqrt {C_{1}} x - C_{1} - 6 x^{2}}}{6}, \ y{\left (x \right )} = - \frac {\sqrt {6} \sqrt {4 \sqrt {2} \sqrt {C_{1}} x - C_{1} - 6 x^{2}}}{6}, \ y{\left (x \right )} = \frac {\sqrt {6} \sqrt {4 \sqrt {2} \sqrt {C_{1}} x - C_{1} - 6 x^{2}}}{6}, \ y{\left (x \right )} = - \frac {\sqrt {6} \sqrt {- 4 \sqrt {2} \sqrt {C_{1}} x - C_{1} - 6 x^{2}}}{6}, \ y{\left (x \right )} = \frac {\sqrt {6} \sqrt {- 4 \sqrt {2} \sqrt {C_{1}} x - C_{1} - 6 x^{2}}}{6}, \ y{\left (x \right )} = - \frac {\sqrt {6} \sqrt {C_{1} - 6 x^{2} + 4 \sqrt {2} x \sqrt {- C_{1}}}}{6}, \ y{\left (x \right )} = \frac {\sqrt {6} \sqrt {C_{1} - 6 x^{2} + 4 \sqrt {2} x \sqrt {- C_{1}}}}{6}, \ y{\left (x \right )} = - \frac {\sqrt {6} \sqrt {- 4 \sqrt {2} \sqrt {C_{1}} x - C_{1} - 6 x^{2}}}{6}, \ y{\left (x \right )} = \frac {\sqrt {6} \sqrt {- 4 \sqrt {2} \sqrt {C_{1}} x - C_{1} - 6 x^{2}}}{6}, \ y{\left (x \right )} = - \frac {\sqrt {6} \sqrt {4 \sqrt {2} \sqrt {C_{1}} x - C_{1} - 6 x^{2}}}{6}, \ y{\left (x \right )} = \frac {\sqrt {6} \sqrt {4 \sqrt {2} \sqrt {C_{1}} x - C_{1} - 6 x^{2}}}{6}\right ]
\]