Internal
problem
ID
[137]
Book
:
Elementary
Differential
Equations.
By
C.
Henry
Edwards,
David
E.
Penney
and
David
Calvis.
6th
edition.
2008
Section
:
Chapter
1.
First
order
differential
equations.
Section
1.6
(substitution
and
exact
equations).
Problems
at
page
72
Problem
number
:
33
Date
solved
:
Friday, February 07, 2025 at 07:56:45 AM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} 3 x^{2}+2 y^{2}+\left (4 x y+6 y^{2}\right ) y^{\prime }&=0 \end{align*}
Time used: 0.038 (sec). Leaf size: 401
\begin{align*}
y &= \frac {\frac {\left (54-62 x^{3} c_1^{3}+6 \sqrt {105 c_1^{6} x^{6}-186 x^{3} c_1^{3}+81}\right )^{{1}/{3}}}{2}+\frac {2 c_1^{2} x^{2}}{\left (54-62 x^{3} c_1^{3}+6 \sqrt {105 c_1^{6} x^{6}-186 x^{3} c_1^{3}+81}\right )^{{1}/{3}}}-c_1 x}{3 c_1} \\
y &= \frac {4 i \sqrt {3}\, c_1^{2} x^{2}-i \left (54-62 x^{3} c_1^{3}+6 \sqrt {105 c_1^{6} x^{6}-186 x^{3} c_1^{3}+81}\right )^{{2}/{3}} \sqrt {3}-4 c_1^{2} x^{2}-4 c_1 x \left (54-62 x^{3} c_1^{3}+6 \sqrt {105 c_1^{6} x^{6}-186 x^{3} c_1^{3}+81}\right )^{{1}/{3}}-\left (54-62 x^{3} c_1^{3}+6 \sqrt {105 c_1^{6} x^{6}-186 x^{3} c_1^{3}+81}\right )^{{2}/{3}}}{12 \left (54-62 x^{3} c_1^{3}+6 \sqrt {105 c_1^{6} x^{6}-186 x^{3} c_1^{3}+81}\right )^{{1}/{3}} c_1} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (54-62 x^{3} c_1^{3}+6 \sqrt {105 c_1^{6} x^{6}-186 x^{3} c_1^{3}+81}\right )^{{2}/{3}}-4 \left (i x c_1 \sqrt {3}+c_1 x +\left (54-62 x^{3} c_1^{3}+6 \sqrt {105 c_1^{6} x^{6}-186 x^{3} c_1^{3}+81}\right )^{{1}/{3}}\right ) c_1 x}{12 \left (54-62 x^{3} c_1^{3}+6 \sqrt {105 c_1^{6} x^{6}-186 x^{3} c_1^{3}+81}\right )^{{1}/{3}} c_1} \\
\end{align*}
Time used: 43.116 (sec). Leaf size: 679
\begin{align*}
y(x)\to \frac {\sqrt [3]{-124 x^3+\sqrt {-256 x^6+\left (-124 x^3+108 e^{2 c_1}\right ){}^2}+108 e^{2 c_1}}}{6 \sqrt [3]{2}}+\frac {2 \sqrt [3]{2} x^2}{3 \sqrt [3]{-124 x^3+\sqrt {-256 x^6+\left (-124 x^3+108 e^{2 c_1}\right ){}^2}+108 e^{2 c_1}}}-\frac {x}{3} \\
y(x)\to \frac {1}{12} i \left (\sqrt {3}+i\right ) \sqrt [3]{-62 x^3+6 \sqrt {3} \sqrt {35 x^6-62 e^{2 c_1} x^3+27 e^{4 c_1}}+54 e^{2 c_1}}-\frac {i \left (\sqrt {3}-i\right ) x^2}{3 \sqrt [3]{-62 x^3+6 \sqrt {3} \sqrt {35 x^6-62 e^{2 c_1} x^3+27 e^{4 c_1}}+54 e^{2 c_1}}}-\frac {x}{3} \\
y(x)\to -\frac {1}{12} i \left (\sqrt {3}-i\right ) \sqrt [3]{-62 x^3+6 \sqrt {3} \sqrt {35 x^6-62 e^{2 c_1} x^3+27 e^{4 c_1}}+54 e^{2 c_1}}+\frac {i \left (\sqrt {3}+i\right ) x^2}{3 \sqrt [3]{-62 x^3+6 \sqrt {3} \sqrt {35 x^6-62 e^{2 c_1} x^3+27 e^{4 c_1}}+54 e^{2 c_1}}}-\frac {x}{3} \\
y(x)\to \frac {1}{6} \left (\sqrt [3]{6 \sqrt {105} \sqrt {x^6}-62 x^3}+\frac {2\ 2^{2/3} x^2}{\sqrt [3]{3 \sqrt {105} \sqrt {x^6}-31 x^3}}-2 x\right ) \\
y(x)\to \frac {1}{12} \left (i \left (\sqrt {3}+i\right ) \sqrt [3]{6 \sqrt {105} \sqrt {x^6}-62 x^3}-\frac {2 i 2^{2/3} \left (\sqrt {3}-i\right ) x^2}{\sqrt [3]{3 \sqrt {105} \sqrt {x^6}-31 x^3}}-4 x\right ) \\
y(x)\to \frac {1}{12} \left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{6 \sqrt {105} \sqrt {x^6}-62 x^3}+\frac {2 i 2^{2/3} \left (\sqrt {3}+i\right ) x^2}{\sqrt [3]{3 \sqrt {105} \sqrt {x^6}-31 x^3}}-4 x\right ) \\
\end{align*}