75.7.7 problem 181
Internal
problem
ID
[16729]
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
:
181
Date
solved
:
Monday, March 31, 2025 at 03:12:53 PM
CAS
classification
:
[_exact, _rational]
\begin{align*} 3 x^{2}-2 x -y+\left (2 y-x +3 y^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 635
ode:=3*x^2-2*x-y(x)+(2*y(x)-x+3*y(x)^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-36 x -108 x^{3}+108 x^{2}-108 c_1 -8+12 \sqrt {81 x^{6}-162 x^{5}+162 c_1 \,x^{3}+135 x^{4}-162 c_1 \,x^{2}-54 x^{3}+81 c_1^{2}+54 c_1 x -15 x^{2}+12 c_1}\right )^{{1}/{3}}}{6}+\frac {2 x +\frac {2}{3}}{\left (-36 x -108 x^{3}+108 x^{2}-108 c_1 -8+12 \sqrt {81 x^{6}-162 x^{5}+162 c_1 \,x^{3}+135 x^{4}-162 c_1 \,x^{2}-54 x^{3}+81 c_1^{2}+54 c_1 x -15 x^{2}+12 c_1}\right )^{{1}/{3}}}-\frac {1}{3} \\
y &= \frac {i \left (4-\left (-36 x -108 x^{3}+108 x^{2}-108 c_1 -8+12 \sqrt {81 x^{6}-162 x^{5}+135 x^{4}+\left (162 c_1 -54\right ) x^{3}+\left (-162 c_1 -15\right ) x^{2}+54 c_1 x +81 c_1^{2}+12 c_1}\right )^{{2}/{3}}+12 x \right ) \sqrt {3}-12 x -{\left (\left (-36 x -108 x^{3}+108 x^{2}-108 c_1 -8+12 \sqrt {81 x^{6}-162 x^{5}+135 x^{4}+\left (162 c_1 -54\right ) x^{3}+\left (-162 c_1 -15\right ) x^{2}+54 c_1 x +81 c_1^{2}+12 c_1}\right )^{{1}/{3}}+2\right )}^{2}}{12 \left (-36 x -108 x^{3}+108 x^{2}-108 c_1 -8+12 \sqrt {81 x^{6}-162 x^{5}+135 x^{4}+\left (162 c_1 -54\right ) x^{3}+\left (-162 c_1 -15\right ) x^{2}+54 c_1 x +81 c_1^{2}+12 c_1}\right )^{{1}/{3}}} \\
y &= \frac {i \left (\left (-36 x -108 x^{3}+108 x^{2}-108 c_1 -8+12 \sqrt {81 x^{6}-162 x^{5}+135 x^{4}+\left (162 c_1 -54\right ) x^{3}+\left (-162 c_1 -15\right ) x^{2}+54 c_1 x +81 c_1^{2}+12 c_1}\right )^{{2}/{3}}-12 x -4\right ) \sqrt {3}-12 x -{\left (\left (-36 x -108 x^{3}+108 x^{2}-108 c_1 -8+12 \sqrt {81 x^{6}-162 x^{5}+135 x^{4}+\left (162 c_1 -54\right ) x^{3}+\left (-162 c_1 -15\right ) x^{2}+54 c_1 x +81 c_1^{2}+12 c_1}\right )^{{1}/{3}}+2\right )}^{2}}{12 \left (-36 x -108 x^{3}+108 x^{2}-108 c_1 -8+12 \sqrt {81 x^{6}-162 x^{5}+135 x^{4}+\left (162 c_1 -54\right ) x^{3}+\left (-162 c_1 -15\right ) x^{2}+54 c_1 x +81 c_1^{2}+12 c_1}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 5.835 (sec). Leaf size: 478
ode=( 3*x^2-2*x-y[x] )+( 2*y[x]-x+3*y[x]^2)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{6} \left (-\frac {2 \sqrt [3]{2} (3 x+1)}{\sqrt [3]{27 x^3-27 x^2+\sqrt {-4 (3 x+1)^3+\left (27 x^3-27 x^2+9 x+2+27 c_1\right ){}^2}+9 x+2+27 c_1}}-2^{2/3} \sqrt [3]{27 x^3-27 x^2+\sqrt {-4 (3 x+1)^3+\left (27 x^3-27 x^2+9 x+2+27 c_1\right ){}^2}+9 x+2+27 c_1}-2\right ) \\
y(x)\to \frac {1}{12} \left (\frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) (3 x+1)}{\sqrt [3]{27 x^3-27 x^2+\sqrt {-4 (3 x+1)^3+\left (27 x^3-27 x^2+9 x+2+27 c_1\right ){}^2}+9 x+2+27 c_1}}+2^{2/3} \left (1-i \sqrt {3}\right ) \sqrt [3]{27 x^3-27 x^2+\sqrt {-4 (3 x+1)^3+\left (27 x^3-27 x^2+9 x+2+27 c_1\right ){}^2}+9 x+2+27 c_1}-4\right ) \\
y(x)\to \frac {1}{12} \left (\frac {2 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) (3 x+1)}{\sqrt [3]{27 x^3-27 x^2+\sqrt {-4 (3 x+1)^3+\left (27 x^3-27 x^2+9 x+2+27 c_1\right ){}^2}+9 x+2+27 c_1}}+2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{27 x^3-27 x^2+\sqrt {-4 (3 x+1)^3+\left (27 x^3-27 x^2+9 x+2+27 c_1\right ){}^2}+9 x+2+27 c_1}-4\right ) \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(3*x**2 - 2*x + (-x + 3*y(x)**2 + 2*y(x))*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out