78.5.1 problem 2 (a)
Internal
problem
ID
[18144]
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.
Section
9
(Integrating
Factors).
Problems
at
page
80
Problem
number
:
2
(a)
Date
solved
:
Tuesday, January 28, 2025 at 11:33:50 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} \left (3 x^{2}-y^{2}\right ) y^{\prime }-2 y x&=0 \end{align*}
✓ Solution by Maple
Time used: 0.010 (sec). Leaf size: 313
dsolve((3*x^2-y(x)^2)*diff(y(x),x) - (2*x*y(x))=0,y(x), singsol=all)
\begin{align*}
y &= \frac {1+\frac {\left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}}}{2}+\frac {2}{\left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}}}}{3 c_1} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{2}/{3}}-4 i \sqrt {3}-4 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}}+4}{12 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}} c_1} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{2}/{3}}-4 i \sqrt {3}+4 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}}-4}{12 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}} c_1} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.167 (sec). Leaf size: 458
DSolve[(3*x^2-y[x]^2)*D[y[x],x] -2*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {1}{3} \left (\frac {\sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2} e^{2 c_1}}{\sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-e^{c_1}\right ) \\
y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{6 \sqrt [3]{2}}-\frac {i \left (\sqrt {3}-i\right ) e^{2 c_1}}{3\ 2^{2/3} \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-\frac {e^{c_1}}{3} \\
y(x)\to -\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{6 \sqrt [3]{2}}+\frac {i \left (\sqrt {3}+i\right ) e^{2 c_1}}{3\ 2^{2/3} \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-\frac {e^{c_1}}{3} \\
\end{align*}