50.6.1 problem 1(a)
Internal
problem
ID
[7893]
Book
:
Differential
Equations:
Theory,
Technique,
and
Practice
by
George
Simmons,
Steven
Krantz.
McGraw-Hill
NY.
2007.
1st
Edition.
Section
:
Chapter
1.
What
is
a
differential
equation.
Section
1.8.
Integrating
Factors.
Page
32
Problem
number
:
1(a)
Date
solved
:
Monday, January 27, 2025 at 03:31:30 PM
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: 0.019 (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: 60.185 (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*}