73.3.32 problem 4.7 (f)
Internal
problem
ID
[15066]
Book
:
Ordinary
Differential
Equations.
An
introduction
to
the
fundamentals.
Kenneth
B.
Howell.
second
edition.
CRC
Press.
FL,
USA.
2020
Section
:
Chapter
4.
SEPARABLE
FIRST
ORDER
EQUATIONS.
Additional
exercises.
page
90
Problem
number
:
4.7
(f)
Date
solved
:
Tuesday, January 28, 2025 at 07:29:19 AM
CAS
classification
:
[_separable]
\begin{align*} y^{\prime }&=\frac {6 x^{2}+4}{3 y^{2}-4 y} \end{align*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 414
dsolve(diff(y(x),x)=(6*x^2+4)/(3*y(x)^2-4*y(x)),y(x), singsol=all)
\begin{align*}
y &= \frac {\left (8+27 x^{3}+27 c_{1} +54 x +3 \sqrt {81}\, \sqrt {\left (x^{3}+c_{1} +2 x \right ) \left (x^{3}+c_{1} +2 x +\frac {16}{27}\right )}\right )^{{2}/{3}}+2 \left (8+27 x^{3}+27 c_{1} +54 x +3 \sqrt {81}\, \sqrt {\left (x^{3}+c_{1} +2 x \right ) \left (x^{3}+c_{1} +2 x +\frac {16}{27}\right )}\right )^{{1}/{3}}+4}{3 \left (8+27 x^{3}+27 c_{1} +54 x +3 \sqrt {81}\, \sqrt {\left (x^{3}+c_{1} +2 x \right ) \left (x^{3}+c_{1} +2 x +\frac {16}{27}\right )}\right )^{{1}/{3}}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (8+27 x^{3}+27 c_{1} +54 x +3 \sqrt {81}\, \sqrt {\left (x^{3}+c_{1} +2 x \right ) \left (x^{3}+c_{1} +2 x +\frac {16}{27}\right )}\right )^{{2}/{3}}-4 i \sqrt {3}-4 \left (8+27 x^{3}+27 c_{1} +54 x +3 \sqrt {81}\, \sqrt {\left (x^{3}+c_{1} +2 x \right ) \left (x^{3}+c_{1} +2 x +\frac {16}{27}\right )}\right )^{{1}/{3}}+4}{6 \left (8+27 x^{3}+27 c_{1} +54 x +3 \sqrt {81}\, \sqrt {\left (x^{3}+c_{1} +2 x \right ) \left (x^{3}+c_{1} +2 x +\frac {16}{27}\right )}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (8+27 x^{3}+27 c_{1} +54 x +3 \sqrt {81}\, \sqrt {\left (x^{3}+c_{1} +2 x \right ) \left (x^{3}+c_{1} +2 x +\frac {16}{27}\right )}\right )^{{2}/{3}}-4 i \sqrt {3}+4 \left (8+27 x^{3}+27 c_{1} +54 x +3 \sqrt {81}\, \sqrt {\left (x^{3}+c_{1} +2 x \right ) \left (x^{3}+c_{1} +2 x +\frac {16}{27}\right )}\right )^{{1}/{3}}-4}{6 \left (8+27 x^{3}+27 c_{1} +54 x +3 \sqrt {81}\, \sqrt {\left (x^{3}+c_{1} +2 x \right ) \left (x^{3}+c_{1} +2 x +\frac {16}{27}\right )}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 3.386 (sec). Leaf size: 356
DSolve[D[y[x],x]==(6*x^2+4)/(3*y[x]^2-4*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {1}{6} \left (2^{2/3} \sqrt [3]{54 x^3+\sqrt {-256+\left (54 x^3+108 x+16+27 c_1\right ){}^2}+108 x+16+27 c_1}+\frac {8 \sqrt [3]{2}}{\sqrt [3]{54 x^3+\sqrt {-256+\left (54 x^3+108 x+16+27 c_1\right ){}^2}+108 x+16+27 c_1}}+4\right ) \\
y(x)\to \frac {1}{12} \left (i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{54 x^3+\sqrt {-256+\left (54 x^3+108 x+16+27 c_1\right ){}^2}+108 x+16+27 c_1}-\frac {8 \sqrt [3]{2} \left (1+i \sqrt {3}\right )}{\sqrt [3]{54 x^3+\sqrt {-256+\left (54 x^3+108 x+16+27 c_1\right ){}^2}+108 x+16+27 c_1}}+8\right ) \\
y(x)\to \frac {1}{12} \left (-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{54 x^3+\sqrt {-256+\left (54 x^3+108 x+16+27 c_1\right ){}^2}+108 x+16+27 c_1}+\frac {8 i \sqrt [3]{2} \left (\sqrt {3}+i\right )}{\sqrt [3]{54 x^3+\sqrt {-256+\left (54 x^3+108 x+16+27 c_1\right ){}^2}+108 x+16+27 c_1}}+8\right ) \\
\end{align*}