78.8.42 problem 42
Internal
problem
ID
[18169]
Book
:
DIFFERENTIAL
EQUATIONS
WITH
APPLICATIONS
AND
HISTORICAL
NOTES
by
George
F.
Simmons.
3rd
edition.
2017.
CRC
press,
Boca
Raton
FL.
Section
:
Chapter
2.
First
order
equations.
Miscellaneous
Problems
for
Chapter
2.
Problems
at
page
99
Problem
number
:
42
Date
solved
:
Monday, March 31, 2025 at 05:20:54 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} 3 x y+y^{2}+\left (3 x y+x^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.233 (sec). Leaf size: 264
ode:=3*x*y(x)+y(x)^2+(3*x*y(x)+x^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {12^{{2}/{3}} \left (12^{{1}/{3}} c_1^{2} x^{2}-{\left (\left (\sqrt {3}\, \sqrt {4 c_1^{4} x^{4}+27}+9\right ) c_1 x \right )}^{{2}/{3}}\right )^{2}}{36 x \,c_1^{2} {\left (\left (\sqrt {3}\, \sqrt {4 c_1^{4} x^{4}+27}+9\right ) c_1 x \right )}^{{2}/{3}}} \\
y &= \frac {{\left (\left (1+i \sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {4 c_1^{4} x^{4}+27}+9\right ) c_1 x \right )}^{{2}/{3}}+c_1^{2} 2^{{2}/{3}} x^{2} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right )\right )}^{2} 3^{{2}/{3}} 2^{{1}/{3}}}{72 {\left (\left (\sqrt {3}\, \sqrt {4 c_1^{4} x^{4}+27}+9\right ) c_1 x \right )}^{{2}/{3}} x \,c_1^{2}} \\
y &= \frac {{\left (\left (i \sqrt {3}-1\right ) {\left (\left (\sqrt {3}\, \sqrt {4 c_1^{4} x^{4}+27}+9\right ) c_1 x \right )}^{{2}/{3}}+c_1^{2} 2^{{2}/{3}} x^{2} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right )\right )}^{2} 3^{{2}/{3}} 2^{{1}/{3}}}{72 {\left (\left (\sqrt {3}\, \sqrt {4 c_1^{4} x^{4}+27}+9\right ) c_1 x \right )}^{{2}/{3}} x \,c_1^{2}} \\
\end{align*}
✓ Mathematica. Time used: 60.144 (sec). Leaf size: 419
ode=(3*x*y[x]+y[x]^2) + (3*x*y[x]+x^2) * D[y[x],x] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{6} \left (2^{2/3} \sqrt [3]{\frac {2 x^4+3 \sqrt {3} \sqrt {e^{c_1} \left (4 x^4+27 e^{c_1}\right )}+27 e^{c_1}}{x}}+\frac {2 \sqrt [3]{2} x^2}{\sqrt [3]{\frac {2 x^4+3 \sqrt {3} \sqrt {e^{c_1} \left (4 x^4+27 e^{c_1}\right )}+27 e^{c_1}}{x}}}-4 x\right ) \\
y(x)\to \frac {1}{12} \left (i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{\frac {2 x^4+3 \sqrt {3} \sqrt {e^{c_1} \left (4 x^4+27 e^{c_1}\right )}+27 e^{c_1}}{x}}-\frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{\frac {2 x^4+3 \sqrt {3} \sqrt {e^{c_1} \left (4 x^4+27 e^{c_1}\right )}+27 e^{c_1}}{x}}}-8 x\right ) \\
y(x)\to \frac {1}{12} \left (-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{\frac {2 x^4+3 \sqrt {3} \sqrt {e^{c_1} \left (4 x^4+27 e^{c_1}\right )}+27 e^{c_1}}{x}}+\frac {2 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) x^2}{\sqrt [3]{\frac {2 x^4+3 \sqrt {3} \sqrt {e^{c_1} \left (4 x^4+27 e^{c_1}\right )}+27 e^{c_1}}{x}}}-8 x\right ) \\
\end{align*}
✓ Sympy. Time used: 95.410 (sec). Leaf size: 430
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(3*x*y(x) + (x**2 + 3*x*y(x))*Derivative(y(x), x) + y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt [3]{2} x^{2}}{3 \sqrt [3]{- \frac {27 C_{1}}{x} - 2 x^{3} + 3 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{x^{2}} + 4 x^{2}\right )}}} - \frac {2 x}{3} - \frac {2^{\frac {2}{3}} \sqrt [3]{- \frac {27 C_{1}}{x} - 2 x^{3} + 3 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{x^{2}} + 4 x^{2}\right )}}}{6}, \ y{\left (x \right )} = \frac {\frac {4 \sqrt [3]{2} x^{2}}{\sqrt [3]{- \frac {27 C_{1}}{x} - 2 x^{3} + 3 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{x^{2}} + 4 x^{2}\right )}}} - 4 x + 4 \sqrt {3} i x - 2^{\frac {2}{3}} \sqrt [3]{- \frac {27 C_{1}}{x} - 2 x^{3} + 3 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{x^{2}} + 4 x^{2}\right )}} - 2^{\frac {2}{3}} \sqrt {3} i \sqrt [3]{- \frac {27 C_{1}}{x} - 2 x^{3} + 3 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{x^{2}} + 4 x^{2}\right )}}}{6 \left (1 - \sqrt {3} i\right )}, \ y{\left (x \right )} = \frac {\frac {4 \sqrt [3]{2} x^{2}}{\sqrt [3]{- \frac {27 C_{1}}{x} - 2 x^{3} + 3 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{x^{2}} + 4 x^{2}\right )}}} - 4 x - 4 \sqrt {3} i x - 2^{\frac {2}{3}} \sqrt [3]{- \frac {27 C_{1}}{x} - 2 x^{3} + 3 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{x^{2}} + 4 x^{2}\right )}} + 2^{\frac {2}{3}} \sqrt {3} i \sqrt [3]{- \frac {27 C_{1}}{x} - 2 x^{3} + 3 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{x^{2}} + 4 x^{2}\right )}}}{6 \left (1 + \sqrt {3} i\right )}\right ]
\]