88.7.1 problem 1
Internal
problem
ID
[23996]
Book
:
Elementary
Differential
Equations.
By
Lee
Roy
Wilcox
and
Herbert
J.
Curtis.
1961
first
edition.
International
texbook
company.
Scranton,
Penn.
USA.
CAT
number
61-15976
Section
:
Chapter
2.
Differential
equations
of
first
order.
Exercise
at
page
41
Problem
number
:
1
Date
solved
:
Thursday, October 02, 2025 at 09:50:25 PM
CAS
classification
:
[_exact, _rational]
\begin{align*} x^{2} y-2 x +\left (y^{2}+\frac {x^{3}}{3}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 326
ode:=x^2*y(x)-2*x+(y(x)^2+1/3*x^3)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (324 x^{2}-324 c_1 +12 \sqrt {12 x^{9}+729 x^{4}-1458 c_1 \,x^{2}+729 c_1^{2}}\right )^{{1}/{3}}}{6}-\frac {2 x^{3}}{\left (324 x^{2}-324 c_1 +12 \sqrt {12 x^{9}+729 x^{4}-1458 c_1 \,x^{2}+729 c_1^{2}}\right )^{{1}/{3}}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (324 x^{2}-324 c_1 +12 \sqrt {12 x^{9}+729 x^{4}-1458 c_1 \,x^{2}+729 c_1^{2}}\right )^{{1}/{3}}}{12}-\frac {\left (i \sqrt {3}-1\right ) x^{3}}{\left (324 x^{2}-324 c_1 +12 \sqrt {12 x^{9}+729 x^{4}-1458 c_1 \,x^{2}+729 c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {12 i \sqrt {3}\, x^{3}+i \sqrt {3}\, \left (324 x^{2}-324 c_1 +12 \sqrt {12 x^{9}+729 x^{4}-1458 c_1 \,x^{2}+729 c_1^{2}}\right )^{{2}/{3}}+12 x^{3}-\left (324 x^{2}-324 c_1 +12 \sqrt {12 x^{9}+729 x^{4}-1458 c_1 \,x^{2}+729 c_1^{2}}\right )^{{2}/{3}}}{12 \left (324 x^{2}-324 c_1 +12 \sqrt {12 x^{9}+729 x^{4}-1458 c_1 \,x^{2}+729 c_1^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 4.123 (sec). Leaf size: 331
ode=(x^2*y[x]-2*x)+( y[x]^2+x^3/3)*D[y[x],{x,1}]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\sqrt [3]{81 x^2+\sqrt {108 x^9+729 \left (3 x^2+c_1\right ){}^2}+27 c_1}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} x^3}{\sqrt [3]{81 x^2+\sqrt {108 x^9+729 \left (3 x^2+c_1\right ){}^2}+27 c_1}}\\ y(x)&\to \frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{81 x^2+\sqrt {108 x^9+729 \left (3 x^2+c_1\right ){}^2}+27 c_1}}{6 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) x^3}{2^{2/3} \sqrt [3]{81 x^2+\sqrt {108 x^9+729 \left (3 x^2+c_1\right ){}^2}+27 c_1}}\\ y(x)&\to \frac {\left (1-i \sqrt {3}\right ) x^3}{2^{2/3} \sqrt [3]{81 x^2+\sqrt {108 x^9+729 \left (3 x^2+c_1\right ){}^2}+27 c_1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{81 x^2+\sqrt {108 x^9+729 \left (3 x^2+c_1\right ){}^2}+27 c_1}}{6 \sqrt [3]{2}} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2*y(x) - 2*x + (x**3/3 + y(x)**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out