28.1.76 problem 79
Internal
problem
ID
[4382]
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
:
79
Date
solved
:
Tuesday, March 04, 2025 at 06:33:44 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} 6 y^{2}-x \left (2 x^{3}+y\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.713 (sec). Leaf size: 193
ode:=6*y(x)^2-x*(2*x^3+y(x))*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= -\frac {x^{3} \left (-x^{3}+\sqrt {x^{3} \left (x^{3}+8 c_{1} \right )}-4 c_{1} \right )}{2 c_{1}} \\
y \left (x \right ) &= \frac {x^{3} \left (x^{3}+\sqrt {x^{3} \left (x^{3}+8 c_{1} \right )}+4 c_{1} \right )}{2 c_{1}} \\
y \left (x \right ) &= -\frac {x^{3} \left (-x^{3}+\sqrt {x^{3} \left (x^{3}+8 c_{1} \right )}-4 c_{1} \right )}{2 c_{1}} \\
y \left (x \right ) &= \frac {x^{3} \left (x^{3}+\sqrt {x^{3} \left (x^{3}+8 c_{1} \right )}+4 c_{1} \right )}{2 c_{1}} \\
y \left (x \right ) &= -\frac {x^{3} \left (-x^{3}+\sqrt {x^{3} \left (x^{3}+8 c_{1} \right )}-4 c_{1} \right )}{2 c_{1}} \\
y \left (x \right ) &= \frac {x^{3} \left (x^{3}+\sqrt {x^{3} \left (x^{3}+8 c_{1} \right )}+4 c_{1} \right )}{2 c_{1}} \\
\end{align*}
✓ Mathematica. Time used: 1.307 (sec). Leaf size: 123
ode=6*y[x]^2-(x*(2*x^3+y[x]))*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to 2 x^3 \left (-1+\frac {2}{1-\frac {4 x^{3/2}}{\sqrt {16 x^3+c_1}}}\right ) \\
y(x)\to 2 x^3 \left (-1+\frac {2}{1+\frac {4 x^{3/2}}{\sqrt {16 x^3+c_1}}}\right ) \\
y(x)\to 0 \\
y(x)\to 2 x^3 \\
y(x)\to \frac {2 \left (\left (x^3\right )^{3/2}-x^{9/2}\right )}{x^{3/2}+\sqrt {x^3}} \\
\end{align*}
✓ Sympy. Time used: 1.411 (sec). Leaf size: 78
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x*(2*x**3 + y(x))*Derivative(y(x), x) + 6*y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {3 C_{1} \sqrt {x^{9} \left (9 C_{1}^{2} x^{3} + 8\right )}}{2} + \frac {x^{3} \left (9 C_{1}^{2} x^{3} + 4\right )}{2}, \ y{\left (x \right )} = \frac {3 C_{1} \sqrt {x^{9} \left (9 C_{1}^{2} x^{3} + 8\right )}}{2} + \frac {x^{3} \left (9 C_{1}^{2} x^{3} + 4\right )}{2}\right ]
\]