40.4.22 problem 23 (b)
Internal
problem
ID
[6662]
Book
:
Schaums
Outline.
Theory
and
problems
of
Differential
Equations,
1st
edition.
Frank
Ayres.
McGraw
Hill
1952
Section
:
Chapter
6.
Equations
of
first
order
and
first
degree
(Linear
equations).
Supplemetary
problems.
Page
39
Problem
number
:
23
(b)
Date
solved
:
Sunday, March 30, 2025 at 11:15:23 AM
CAS
classification
:
[_rational]
\begin{align*} 4 x^{2} y y^{\prime }&=3 x \left (3 y^{2}+2\right )+2 \left (3 y^{2}+2\right )^{3} \end{align*}
✓ Maple. Time used: 0.207 (sec). Leaf size: 175
ode:=4*x^2*y(x)*diff(y(x),x) = 3*x*(3*y(x)^2+2)+2*(3*y(x)^2+2)^3;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {\sqrt {\frac {-18 c_1 \,x^{8}-6 \sqrt {-3 \left (c_1 \,x^{8}-\frac {1}{3}\right ) c_1 \,x^{9}}+6}{3 c_1 \,x^{8}-1}}}{3} \\
y &= \frac {\sqrt {\frac {-18 c_1 \,x^{8}-6 \sqrt {-3 \left (c_1 \,x^{8}-\frac {1}{3}\right ) c_1 \,x^{9}}+6}{3 c_1 \,x^{8}-1}}}{3} \\
y &= -\frac {\sqrt {6}\, \sqrt {\frac {-3 c_1 \,x^{8}+\sqrt {-3 \left (c_1 \,x^{8}-\frac {1}{3}\right ) c_1 \,x^{9}}+1}{3 c_1 \,x^{8}-1}}}{3} \\
y &= \frac {\sqrt {6}\, \sqrt {\frac {-3 c_1 \,x^{8}+\sqrt {-3 \left (c_1 \,x^{8}-\frac {1}{3}\right ) c_1 \,x^{9}}+1}{3 c_1 \,x^{8}-1}}}{3} \\
\end{align*}
✓ Mathematica. Time used: 22.093 (sec). Leaf size: 277
ode=4*x^2*y[x]*D[y[x],x]==3*x*(3*y[x]^2+2)+2*(3*y[x]^2+2)^3;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {1}{3} \sqrt {2} \sqrt {-\frac {3 x^8+\sqrt {3} \sqrt {-x^9 \left (x^8+72 c_1\right )}+216 c_1}{x^8+72 c_1}} \\
y(x)\to \frac {1}{3} \sqrt {2} \sqrt {-\frac {3 x^8+\sqrt {3} \sqrt {-x^9 \left (x^8+72 c_1\right )}+216 c_1}{x^8+72 c_1}} \\
y(x)\to -\frac {1}{3} \sqrt {2} \sqrt {\frac {-3 x^8+\sqrt {3} \sqrt {-x^9 \left (x^8+72 c_1\right )}-216 c_1}{x^8+72 c_1}} \\
y(x)\to \frac {1}{3} \sqrt {2} \sqrt {\frac {-3 x^8+\sqrt {3} \sqrt {-x^9 \left (x^8+72 c_1\right )}-216 c_1}{x^8+72 c_1}} \\
y(x)\to -i \sqrt {\frac {2}{3}} \\
y(x)\to i \sqrt {\frac {2}{3}} \\
\end{align*}
✓ Sympy. Time used: 14.755 (sec). Leaf size: 194
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(4*x**2*y(x)*Derivative(y(x), x) - 3*x*(3*y(x)**2 + 2) - 2*(3*y(x)**2 + 2)**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- \frac {48 C_{1} - 3 x^{8} - \sqrt {3} \sqrt {x^{9} \left (16 C_{1} - x^{8}\right )}}{16 C_{1} - x^{8}}}}{3}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- \frac {48 C_{1} - 3 x^{8} - \sqrt {3} \sqrt {x^{9} \left (16 C_{1} - x^{8}\right )}}{16 C_{1} - x^{8}}}}{3}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- \frac {48 C_{1} - 3 x^{8} + \sqrt {3} \sqrt {x^{9} \left (16 C_{1} - x^{8}\right )}}{16 C_{1} - x^{8}}}}{3}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- \frac {48 C_{1} - 3 x^{8} + \sqrt {3} \sqrt {x^{9} \left (16 C_{1} - x^{8}\right )}}{16 C_{1} - x^{8}}}}{3}\right ]
\]