15.8.15 problem 15
Internal
problem
ID
[3018]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
12,
page
46
Problem
number
:
15
Date
solved
:
Sunday, March 30, 2025 at 01:09:14 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} 2 x y+y^{4}+\left (x y^{3}-2 x^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.038 (sec). Leaf size: 326
ode:=2*x*y(x)+y(x)^4+(x*y(x)^3-2*x^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\frac {\left (-108 x^{4}+8 c_1^{3}+12 \sqrt {81 x^{4}-12 c_1^{3}}\, x^{2}\right )^{{1}/{3}}}{2}+\frac {2 c_1^{2}}{\left (-108 x^{4}+8 c_1^{3}+12 \sqrt {81 x^{4}-12 c_1^{3}}\, x^{2}\right )^{{1}/{3}}}+c_1}{3 x} \\
y &= \frac {\left (-i \sqrt {3}-1\right ) \left (-108 x^{4}+8 c_1^{3}+12 \sqrt {81 x^{4}-12 c_1^{3}}\, x^{2}\right )^{{2}/{3}}+4 c_1 \left (i c_1 \sqrt {3}-c_1 +\left (-108 x^{4}+8 c_1^{3}+12 \sqrt {81 x^{4}-12 c_1^{3}}\, x^{2}\right )^{{1}/{3}}\right )}{12 \left (-108 x^{4}+8 c_1^{3}+12 \sqrt {81 x^{4}-12 c_1^{3}}\, x^{2}\right )^{{1}/{3}} x} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (-108 x^{4}+8 c_1^{3}+12 \sqrt {81 x^{4}-12 c_1^{3}}\, x^{2}\right )^{{2}/{3}}+4 \left (-i c_1 \sqrt {3}-c_1 +\left (-108 x^{4}+8 c_1^{3}+12 \sqrt {81 x^{4}-12 c_1^{3}}\, x^{2}\right )^{{1}/{3}}\right ) c_1}{12 \left (-108 x^{4}+8 c_1^{3}+12 \sqrt {81 x^{4}-12 c_1^{3}}\, x^{2}\right )^{{1}/{3}} x} \\
\end{align*}
✓ Mathematica. Time used: 17.179 (sec). Leaf size: 371
ode=(2*x*y[x]+y[x]^4)+(x*y[x]^3-2*x^2)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\frac {2 \sqrt [3]{2} c_1{}^2}{\sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}}+2^{2/3} \sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}+2 c_1}{6 x} \\
y(x)\to \frac {\frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) c_1{}^2}{\sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}}+2^{2/3} \left (1-i \sqrt {3}\right ) \sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}-4 c_1}{12 x} \\
y(x)\to \frac {\frac {2 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) c_1{}^2}{\sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}}+2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}-4 c_1}{12 x} \\
y(x)\to 0 \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x*y(x) + (-2*x**2 + x*y(x)**3)*Derivative(y(x), x) + y(x)**4,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out