10.6.7 problem 7
Internal
problem
ID
[1224]
Book
:
Elementary
differential
equations
and
boundary
value
problems,
10th
ed.,
Boyce
and
DiPrima
Section
:
Miscellaneous
problems,
end
of
chapter
2.
Page
133
Problem
number
:
7
Date
solved
:
Monday, January 27, 2025 at 04:45:55 AM
CAS
classification
:
[_separable]
\begin{align*} y^{\prime }&=\frac {4 x^{3}+1}{y \left (2+3 y\right )} \end{align*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 378
dsolve(diff(y(x),x) = (4*x^3+1)/(y(x)*(2+3*y(x))),y(x), singsol=all)
\begin{align*}
y &= \frac {\left (-8+108 x^{4}+108 c_1 +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_1 +x \right ) \left (x^{4}+c_1 +x -\frac {4}{27}\right )}\right )^{{2}/{3}}-2 \left (-8+108 x^{4}+108 c_1 +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_1 +x \right ) \left (x^{4}+c_1 +x -\frac {4}{27}\right )}\right )^{{1}/{3}}+4}{6 \left (-8+108 x^{4}+108 c_1 +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_1 +x \right ) \left (x^{4}+c_1 +x -\frac {4}{27}\right )}\right )^{{1}/{3}}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (-8+108 x^{4}+108 c_1 +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_1 +x \right ) \left (x^{4}+c_1 +x -\frac {4}{27}\right )}\right )^{{2}/{3}}-4 i \sqrt {3}+4 \left (-8+108 x^{4}+108 c_1 +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_1 +x \right ) \left (x^{4}+c_1 +x -\frac {4}{27}\right )}\right )^{{1}/{3}}+4}{12 \left (-8+108 x^{4}+108 c_1 +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_1 +x \right ) \left (x^{4}+c_1 +x -\frac {4}{27}\right )}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (-8+108 x^{4}+108 c_1 +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_1 +x \right ) \left (x^{4}+c_1 +x -\frac {4}{27}\right )}\right )^{{2}/{3}}-4 i \sqrt {3}-4 \left (-8+108 x^{4}+108 c_1 +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_1 +x \right ) \left (x^{4}+c_1 +x -\frac {4}{27}\right )}\right )^{{1}/{3}}-4}{12 \left (-8+108 x^{4}+108 c_1 +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_1 +x \right ) \left (x^{4}+c_1 +x -\frac {4}{27}\right )}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 4.529 (sec). Leaf size: 356
DSolve[D[y[x],x]== (4*x^3+1)/(y[x]*(2+3*y[x])),y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {1}{6} \left (2^{2/3} \sqrt [3]{27 x^4+\sqrt {-4+\left (27 x^4+27 x-2+27 c_1\right ){}^2}+27 x-2+27 c_1}+\frac {2 \sqrt [3]{2}}{\sqrt [3]{27 x^4+\sqrt {-4+\left (27 x^4+27 x-2+27 c_1\right ){}^2}+27 x-2+27 c_1}}-2\right ) \\
y(x)\to \frac {1}{12} \left (i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{27 x^4+\sqrt {-4+\left (27 x^4+27 x-2+27 c_1\right ){}^2}+27 x-2+27 c_1}-\frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right )}{\sqrt [3]{27 x^4+\sqrt {-4+\left (27 x^4+27 x-2+27 c_1\right ){}^2}+27 x-2+27 c_1}}-4\right ) \\
y(x)\to \frac {1}{12} \left (-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{27 x^4+\sqrt {-4+\left (27 x^4+27 x-2+27 c_1\right ){}^2}+27 x-2+27 c_1}+\frac {2 i \sqrt [3]{2} \left (\sqrt {3}+i\right )}{\sqrt [3]{27 x^4+\sqrt {-4+\left (27 x^4+27 x-2+27 c_1\right ){}^2}+27 x-2+27 c_1}}-4\right ) \\
\end{align*}