22.1.97 problem 120
Internal
problem
ID
[4403]
Book
:
Differential
equations
for
engineers
by
Wei-Chau
XIE,
Cambridge
Press
2010
Section
:
Chapter
2.
First-Order
and
Simple
Higher-Order
Differential
Equations.
Page
78
Problem
number
:
120
Date
solved
:
Tuesday, September 30, 2025 at 07:28:02 AM
CAS
classification
:
[_rational]
\begin{align*} 2 x^{3} y^{2}-y+\left (2 x^{2} y^{3}-x \right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.010 (sec). Leaf size: 357
ode:=2*x^3*y(x)^2-y(x)+(2*x^2*y(x)^3-x)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {12^{{1}/{3}} \left (-{\left (\left (-9+\sqrt {12 x^{8}-36 c_1 \,x^{6}+36 c_1^{2} x^{4}-12 c_1^{3} x^{2}+81}\right ) x^{2}\right )}^{{2}/{3}}+12^{{1}/{3}} x^{2} \left (x^{2}-c_1 \right )\right )}{6 {\left (\left (-9+\sqrt {12 x^{8}-36 c_1 \,x^{6}+36 c_1^{2} x^{4}-12 c_1^{3} x^{2}+81}\right ) x^{2}\right )}^{{1}/{3}} x} \\
y &= -\frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (1+i \sqrt {3}\right ) {\left (\left (-9+\sqrt {12 x^{8}-36 c_1 \,x^{6}+36 c_1^{2} x^{4}-12 c_1^{3} x^{2}+81}\right ) x^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} \left (x^{2}-c_1 \right ) x^{2} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right )\right )}{12 {\left (\left (-9+\sqrt {12 x^{8}-36 c_1 \,x^{6}+36 c_1^{2} x^{4}-12 c_1^{3} x^{2}+81}\right ) x^{2}\right )}^{{1}/{3}} x} \\
y &= \frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (i \sqrt {3}-1\right ) {\left (\left (-9+\sqrt {12 x^{8}-36 c_1 \,x^{6}+36 c_1^{2} x^{4}-12 c_1^{3} x^{2}+81}\right ) x^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) \left (x^{2}-c_1 \right ) x^{2}\right )}{12 {\left (\left (-9+\sqrt {12 x^{8}-36 c_1 \,x^{6}+36 c_1^{2} x^{4}-12 c_1^{3} x^{2}+81}\right ) x^{2}\right )}^{{1}/{3}} x} \\
\end{align*}
✓ Mathematica. Time used: 51.229 (sec). Leaf size: 364
ode=(2*x^2*y[x]^2-y[x])+(2*x^2*y[x]^3-x)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\sqrt [3]{-27 x^2+\sqrt {729 x^4+4 \left (6 x^3-3 c_1 x^2\right ){}^3}}}{3 \sqrt [3]{2} x}-\frac {\sqrt [3]{2} x (2 x-c_1)}{\sqrt [3]{-27 x^2+\sqrt {729 x^4+4 \left (6 x^3-3 c_1 x^2\right ){}^3}}}\\ y(x)&\to \frac {\left (1+i \sqrt {3}\right ) x (2 x-c_1)}{2^{2/3} \sqrt [3]{-27 x^2+\sqrt {729 x^4+4 \left (6 x^3-3 c_1 x^2\right ){}^3}}}+\frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{-27 x^2+\sqrt {729 x^4+4 \left (6 x^3-3 c_1 x^2\right ){}^3}}}{6 \sqrt [3]{2} x}\\ y(x)&\to \frac {\left (1-i \sqrt {3}\right ) x (2 x-c_1)}{2^{2/3} \sqrt [3]{-27 x^2+\sqrt {729 x^4+4 \left (6 x^3-3 c_1 x^2\right ){}^3}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-27 x^2+\sqrt {729 x^4+4 \left (6 x^3-3 c_1 x^2\right ){}^3}}}{6 \sqrt [3]{2} x} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x**3*y(x)**2 + (2*x**2*y(x)**3 - x)*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out