66.1.40 problem Problem 54
Internal
problem
ID
[13816]
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
54
Date
solved
:
Monday, March 31, 2025 at 08:13:51 AM
CAS
classification
:
[_rational]
\begin{align*} x^{2}-y+\left (x^{2} y^{2}+x \right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 342
ode:=x^2-y(x)+(x^2*y(x)^2+x)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {\left (-\frac {2^{{1}/{3}} {\left (\left (-3 c_1 x -3 x^{2}+\sqrt {\frac {9 c_1^{2} x^{3}+18 c_1 \,x^{4}+9 x^{5}+4}{x}}\right ) x^{2}\right )}^{{2}/{3}}}{2}+x \right ) 2^{{1}/{3}}}{{\left (\left (-3 c_1 x -3 x^{2}+\sqrt {\frac {9 c_1^{2} x^{3}+18 c_1 \,x^{4}+9 x^{5}+4}{x}}\right ) x^{2}\right )}^{{1}/{3}} x} \\
y &= -\frac {\left (\left (1+i \sqrt {3}\right ) 2^{{1}/{3}} {\left (\left (-3 c_1 x -3 x^{2}+\sqrt {\frac {9 c_1^{2} x^{3}+18 c_1 \,x^{4}+9 x^{5}+4}{x}}\right ) x^{2}\right )}^{{2}/{3}}+2 i \sqrt {3}\, x -2 x \right ) 2^{{1}/{3}}}{4 {\left (\left (-3 c_1 x -3 x^{2}+\sqrt {\frac {9 c_1^{2} x^{3}+18 c_1 \,x^{4}+9 x^{5}+4}{x}}\right ) x^{2}\right )}^{{1}/{3}} x} \\
y &= \frac {\left (i \sqrt {3}-1\right ) {\left (\left (-3 c_1 x -3 x^{2}+\sqrt {\frac {9 c_1^{2} x^{3}+18 c_1 \,x^{4}+9 x^{5}+4}{x}}\right ) x^{2}\right )}^{{2}/{3}} 2^{{2}/{3}}+2 \left (1+i \sqrt {3}\right ) x 2^{{1}/{3}}}{4 {\left (\left (-3 c_1 x -3 x^{2}+\sqrt {\frac {9 c_1^{2} x^{3}+18 c_1 \,x^{4}+9 x^{5}+4}{x}}\right ) x^{2}\right )}^{{1}/{3}} x} \\
\end{align*}
✓ Mathematica. Time used: 60.079 (sec). Leaf size: 400
ode=(x^2-y[x])+(x^2*y[x]^2+x)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {-2 \sqrt [3]{2} x+\left (-6 x^4+6 c_1 x^3+2 \sqrt {x^3 \left (9 x^5-18 c_1 x^4+9 c_1{}^2 x^3+4\right )}\right ){}^{2/3}}{2 x \sqrt [3]{-3 x^4+3 c_1 x^3+\sqrt {x^3 \left (9 x^5-18 c_1 x^4+9 c_1{}^2 x^3+4\right )}}} \\
y(x)\to \frac {i \left (\sqrt {3}+i\right ) \left (-6 x^4+6 c_1 x^3+2 \sqrt {x^3 \left (9 x^5-18 c_1 x^4+9 c_1{}^2 x^3+4\right )}\right ){}^{2/3}+\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x}{4 x \sqrt [3]{-3 x^4+3 c_1 x^3+\sqrt {x^3 \left (9 x^5-18 c_1 x^4+9 c_1{}^2 x^3+4\right )}}} \\
y(x)\to \frac {\left (-1-i \sqrt {3}\right ) \left (-6 x^4+6 c_1 x^3+2 \sqrt {x^3 \left (9 x^5-18 c_1 x^4+9 c_1{}^2 x^3+4\right )}\right ){}^{2/3}+\sqrt [3]{2} \left (2-2 i \sqrt {3}\right ) x}{4 x \sqrt [3]{-3 x^4+3 c_1 x^3+\sqrt {x^3 \left (9 x^5-18 c_1 x^4+9 c_1{}^2 x^3+4\right )}}} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2 + (x**2*y(x)**2 + x)*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out