78.8.42 problem 42
Internal
problem
ID
[18240]
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
:
Tuesday, January 28, 2025 at 11:43:22 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} 3 y x +y^{2}+\left (3 y x +x^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.116 (sec). Leaf size: 264
dsolve((3*x*y(x)+y(x)^2) + (3*x*y(x)+x^2) * diff(y(x),x) = 0,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 {2^{{1}/{3}} {\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} x^{2} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) 2^{{2}/{3}}\right )}^{2} 3^{{2}/{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 {2^{{1}/{3}} 3^{{2}/{3}} {\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} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) x^{2} 2^{{2}/{3}}\right )}^{2}}{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*}
✓ Solution by Mathematica
Time used: 60.144 (sec). Leaf size: 419
DSolve[(3*x*y[x]+y[x]^2) + (3*x*y[x]+x^2) * D[y[x],x] == 0,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*}