69.1.57 problem 76
Internal
problem
ID
[14139]
Book
:
DIFFERENTIAL
and
INTEGRAL
CALCULUS.
VOL
I.
by
N.
PISKUNOV.
MIR
PUBLISHERS,
Moscow
1969.
Section
:
Chapter
8.
Differential
equations.
Exercises
page
595
Problem
number
:
76
Date
solved
:
Monday, March 31, 2025 at 12:10:22 PM
CAS
classification
:
[_exact, _rational]
\begin{align*} 6 x y^{2}+4 x^{3}+3 \left (2 x^{2} y+y^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 418
ode:=6*x*y(x)^2+4*x^3+3*(2*x^2*y(x)+y(x)^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-4 x^{4}-4 c_1 -8 x^{6}+4 \sqrt {4 x^{10}+x^{8}+4 c_1 \,x^{6}+2 c_1 \,x^{4}+c_1^{2}}\right )^{{1}/{3}}}{2}+\frac {2 x^{4}}{\left (-4 x^{4}-4 c_1 -8 x^{6}+4 \sqrt {4 x^{10}+x^{8}+4 c_1 \,x^{6}+2 c_1 \,x^{4}+c_1^{2}}\right )^{{1}/{3}}}-x^{2} \\
y &= \frac {4 i \sqrt {3}\, x^{4}-i \sqrt {3}\, \left (-4 x^{4}-4 c_1 -8 x^{6}+4 \sqrt {\left (4 x^{6}+x^{4}+c_1 \right ) \left (x^{4}+c_1 \right )}\right )^{{2}/{3}}-4 x^{4}-4 x^{2} \left (-4 x^{4}-4 c_1 -8 x^{6}+4 \sqrt {\left (4 x^{6}+x^{4}+c_1 \right ) \left (x^{4}+c_1 \right )}\right )^{{1}/{3}}-\left (-4 x^{4}-4 c_1 -8 x^{6}+4 \sqrt {\left (4 x^{6}+x^{4}+c_1 \right ) \left (x^{4}+c_1 \right )}\right )^{{2}/{3}}}{4 \left (-4 x^{4}-4 c_1 -8 x^{6}+4 \sqrt {\left (4 x^{6}+x^{4}+c_1 \right ) \left (x^{4}+c_1 \right )}\right )^{{1}/{3}}} \\
y &= \frac {\left (-4 x^{4}-4 c_1 -8 x^{6}+4 \sqrt {\left (4 x^{6}+x^{4}+c_1 \right ) \left (x^{4}+c_1 \right )}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{4}-\frac {\left (i x^{2} \sqrt {3}+x^{2}+\left (-4 x^{4}-4 c_1 -8 x^{6}+4 \sqrt {\left (4 x^{6}+x^{4}+c_1 \right ) \left (x^{4}+c_1 \right )}\right )^{{1}/{3}}\right ) x^{2}}{\left (-4 x^{4}-4 c_1 -8 x^{6}+4 \sqrt {\left (4 x^{6}+x^{4}+c_1 \right ) \left (x^{4}+c_1 \right )}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 15.392 (sec). Leaf size: 419
ode=2*(3*x*y[x]^2+2*x^3)+3*(2*x^2*y[x]+y[x]^2)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -x^2+\frac {\sqrt [3]{2} x^4}{\sqrt [3]{-2 x^6-x^4+\sqrt {4 x^{10}+x^8-4 c_1 x^6-2 c_1 x^4+c_1{}^2}+c_1}}+\frac {\sqrt [3]{-2 x^6-x^4+\sqrt {4 x^{10}+x^8-4 c_1 x^6-2 c_1 x^4+c_1{}^2}+c_1}}{\sqrt [3]{2}} \\
y(x)\to \frac {1}{4} \left (-4 x^2-\frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) x^4}{\sqrt [3]{-2 x^6-x^4+\sqrt {4 x^{10}+x^8-4 c_1 x^6-2 c_1 x^4+c_1{}^2}+c_1}}+i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{-2 x^6-x^4+\sqrt {4 x^{10}+x^8-4 c_1 x^6-2 c_1 x^4+c_1{}^2}+c_1}\right ) \\
y(x)\to \frac {1}{4} \left (-4 x^2+\frac {2 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) x^4}{\sqrt [3]{-2 x^6-x^4+\sqrt {4 x^{10}+x^8-4 c_1 x^6-2 c_1 x^4+c_1{}^2}+c_1}}+2^{2/3} \left (-1-i \sqrt {3}\right ) \sqrt [3]{-2 x^6-x^4+\sqrt {4 x^{10}+x^8-4 c_1 x^6-2 c_1 x^4+c_1{}^2}+c_1}\right ) \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(4*x**3 + 6*x*y(x)**2 + (6*x**2*y(x) + 3*y(x)**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out