66.1.41 problem Problem 55
Internal
problem
ID
[13817]
Book
:
Differential
equations
and
the
calculus
of
variations
by
L.
ElSGOLTS.
MIR
PUBLISHERS,
MOSCOW,
Third
printing
1977.
Section
:
Chapter
1,
First-Order
Differential
Equations.
Problems
page
88
Problem
number
:
Problem
55
Date
solved
:
Monday, March 31, 2025 at 08:13:53 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} 3 y^{2}-x +2 y \left (y^{2}-3 x \right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.051 (sec). Leaf size: 101
ode:=3*y(x)^2-x+2*y(x)*(y(x)^2-3*x)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {\sqrt {-2 \sqrt {c_1 \left (c_1 -8 x \right )}+2 c_1 -4 x}}{2} \\
y &= \frac {\sqrt {-2 \sqrt {c_1 \left (c_1 -8 x \right )}+2 c_1 -4 x}}{2} \\
y &= -\frac {\sqrt {2 \sqrt {c_1 \left (c_1 -8 x \right )}+2 c_1 -4 x}}{2} \\
y &= \frac {\sqrt {2 \sqrt {c_1 \left (c_1 -8 x \right )}+2 c_1 -4 x}}{2} \\
\end{align*}
✓ Mathematica. Time used: 9.496 (sec). Leaf size: 185
ode=(3*y[x]^2-x)+(2*y[x])*(y[x]^2-3*x)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sqrt {-2 x-e^{\frac {c_1}{2}} \sqrt {8 x+e^{c_1}}-e^{c_1}}}{\sqrt {2}} \\
y(x)\to \frac {\sqrt {-2 x-e^{\frac {c_1}{2}} \sqrt {8 x+e^{c_1}}-e^{c_1}}}{\sqrt {2}} \\
y(x)\to -\frac {\sqrt {-2 x+e^{\frac {c_1}{2}} \sqrt {8 x+e^{c_1}}-e^{c_1}}}{\sqrt {2}} \\
y(x)\to \frac {\sqrt {-2 x+e^{\frac {c_1}{2}} \sqrt {8 x+e^{c_1}}-e^{c_1}}}{\sqrt {2}} \\
\end{align*}
✓ Sympy. Time used: 12.338 (sec). Leaf size: 160
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x + 2*(-3*x + y(x)**2)*y(x)*Derivative(y(x), x) + 3*y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 2 x e^{2 C_{1}} - \sqrt {- 8 x e^{2 C_{1}} + 1} + 1} e^{- C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 2 x e^{2 C_{1}} + \sqrt {- 8 x e^{2 C_{1}} + 1} + 1} e^{- C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 2 x e^{2 C_{1}} + \sqrt {- 8 x e^{2 C_{1}} + 1} + 1} e^{- C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 2 x e^{2 C_{1}} - \sqrt {- 8 x e^{2 C_{1}} + 1} + 1} e^{- C_{1}}}{2}\right ]
\]