85.33.64 problem 65
Internal
problem
ID
[22687]
Book
:
Applied
Differential
Equations.
By
Murray
R.
Spiegel.
3rd
edition.
1980.
Pearson.
ISBN
978-0130400970
Section
:
Chapter
two.
First
order
and
simple
higher
order
ordinary
differential
equations.
A
Exercises
at
page
65
Problem
number
:
65
Date
solved
:
Thursday, October 02, 2025 at 09:06:36 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} x y^{\prime }-y&=2 x^{2} y^{2} y^{\prime } \end{align*}
✓ Maple. Time used: 0.153 (sec). Leaf size: 260
ode:=x*diff(y(x),x)-y(x) = 2*x^2*y(x)^2*diff(y(x),x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {2^{{1}/{3}} \left (c_1^{2} 2^{{1}/{3}} x +{\left (\left (x +\sqrt {\frac {x^{3}-2 c_1^{2}}{x}}\right ) c_1^{2} x^{2}\right )}^{{2}/{3}}\right )}{2 c_1 x {\left (\left (x +\sqrt {\frac {x^{3}-2 c_1^{2}}{x}}\right ) c_1^{2} x^{2}\right )}^{{1}/{3}}} \\
y &= \frac {\left (\left (-i \sqrt {3}-1\right ) {\left (\left (x +\sqrt {\frac {x^{3}-2 c_1^{2}}{x}}\right ) c_1^{2} x^{2}\right )}^{{2}/{3}}+x 2^{{1}/{3}} c_1^{2} \left (i \sqrt {3}-1\right )\right ) 2^{{1}/{3}}}{4 {\left (\left (x +\sqrt {\frac {x^{3}-2 c_1^{2}}{x}}\right ) c_1^{2} x^{2}\right )}^{{1}/{3}} c_1 x} \\
y &= -\frac {\left (\left (1-i \sqrt {3}\right ) {\left (\left (x +\sqrt {\frac {x^{3}-2 c_1^{2}}{x}}\right ) c_1^{2} x^{2}\right )}^{{2}/{3}}+x \left (1+i \sqrt {3}\right ) 2^{{1}/{3}} c_1^{2}\right ) 2^{{1}/{3}}}{4 {\left (\left (x +\sqrt {\frac {x^{3}-2 c_1^{2}}{x}}\right ) c_1^{2} x^{2}\right )}^{{1}/{3}} c_1 x} \\
\end{align*}
✓ Mathematica. Time used: 38.354 (sec). Leaf size: 306
ode=x*D[y[x],x]-y[x]==2*x^2*y[x]^2*D[y[x],x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\sqrt [3]{2}+\frac {\left (3 c_1 x^3+\sqrt {x^3 \left (-2+9 c_1{}^2 x^3\right )}\right ){}^{2/3}}{x}}{2^{2/3} \sqrt [3]{3 c_1 x^3+\sqrt {x^3 \left (-2+9 c_1{}^2 x^3\right )}}}\\ y(x)&\to \frac {i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (3 c_1 x^3+\sqrt {x^3 \left (-2+9 c_1{}^2 x^3\right )}\right ){}^{2/3}+2^{2/3} \left (-1-i \sqrt {3}\right ) x}{4 x \sqrt [3]{3 c_1 x^3+\sqrt {x^3 \left (-2+9 c_1{}^2 x^3\right )}}}\\ y(x)&\to \frac {i \left (\sqrt [3]{2} \left (\sqrt {3}+i\right ) x-\left (\sqrt {3}-i\right ) \left (3 c_1 x^3+\sqrt {x^3 \left (-2+9 c_1{}^2 x^3\right )}\right ){}^{2/3}\right )}{2\ 2^{2/3} x \sqrt [3]{3 c_1 x^3+\sqrt {x^3 \left (-2+9 c_1{}^2 x^3\right )}}}\\ y(x)&\to 0 \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-2*x**2*y(x)**2*Derivative(y(x), x) + x*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out