75.7.21 problem 196
Internal
problem
ID
[16735]
Book
:
A
book
of
problems
in
ordinary
differential
equations.
M.L.
KRASNOV,
A.L.
KISELYOV,
G.I.
MARKARENKO.
MIR,
MOSCOW.
1983
Section
:
Section
7,
Total
differential
equations.
The
integrating
factor.
Exercises
page
61
Problem
number
:
196
Date
solved
:
Thursday, March 13, 2025 at 08:39:40 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} 3 y^{2}-x +\left (2 y^{3}-6 x y\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.485 (sec). Leaf size: 101
ode:=3*y(x)^2-x+(2*y(x)^3-6*x*y(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: 0.192 (sec). Leaf size: 112
ode=( 3*y[x]^2-x)+( 2*y[x]^3-6*x*y[x] )*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^x\left (\frac {2}{y(x)^2+K[1]}-\frac {1}{K[1]-y(x)^2}\right )dK[1]+\int _1^{y(x)}\left (-\frac {2 K[2]}{K[2]^2-x}+\frac {4 K[2]}{K[2]^2+x}-\int _1^x\left (-\frac {2 K[2]}{\left (K[1]-K[2]^2\right )^2}-\frac {4 K[2]}{\left (K[2]^2+K[1]\right )^2}\right )dK[1]\right )dK[2]=c_1,y(x)\right ]
\]
✓ Sympy. Time used: 12.514 (sec). Leaf size: 160
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x + (-6*x*y(x) + 2*y(x)**3)*Derivative(y(x), x) + 3*y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 2 x e^{2 C_{1}} - \sqrt {- 8 x e^{2 C_{1}} + 1} + 1} e^{- C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 2 x e^{2 C_{1}} + \sqrt {- 8 x e^{2 C_{1}} + 1} + 1} e^{- C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 2 x e^{2 C_{1}} + \sqrt {- 8 x e^{2 C_{1}} + 1} + 1} e^{- C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 2 x e^{2 C_{1}} - \sqrt {- 8 x e^{2 C_{1}} + 1} + 1} e^{- C_{1}}}{2}\right ]
\]