49.22.11 problem 2(c)
Internal
problem
ID
[7757]
Book
:
An
introduction
to
Ordinary
Differential
Equations.
Earl
A.
Coddington.
Dover.
NY
1961
Section
:
Chapter
5.
Existence
and
uniqueness
of
solutions
to
first
order
equations.
Page
198
Problem
number
:
2(c)
Date
solved
:
Monday, January 27, 2025 at 03:12:07 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} 5 x^{3} y^{2}+2 y+\left (3 x^{4} y+2 x \right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.330 (sec). Leaf size: 346
dsolve((5*x^3*y(x)^2+2*y(x))+(3*x^4*y(x)+2*x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y &= \frac {\frac {12^{{2}/{3}} \left (12^{{1}/{3}} c_{1}^{2}+{\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{{2}/{3}}\right )^{2}}{36 c_{1}^{2} {\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{{2}/{3}}}-1}{x^{3}} \\
y &= \frac {-\frac {c_{1} {\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{{2}/{3}}}{3}+\frac {3 \,2^{{1}/{3}} \left (i 3^{{1}/{6}}-\frac {3^{{2}/{3}}}{3}\right ) \left (x^{2}+\frac {\sqrt {-12 c_{1}^{4}+81 x^{4}}}{9}\right ) {\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{{1}/{3}}}{4}-\frac {2^{{2}/{3}} c_{1}^{3} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right )}{6}}{c_{1} {\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{{2}/{3}} x^{3}} \\
y &= -\frac {3 \left (\frac {4 c_{1} {\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{{2}/{3}}}{9}+2^{{1}/{3}} \left (x^{2}+\frac {\sqrt {-12 c_{1}^{4}+81 x^{4}}}{9}\right ) \left (i 3^{{1}/{6}}+\frac {3^{{2}/{3}}}{3}\right ) {\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{{1}/{3}}-\frac {2 \,2^{{2}/{3}} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) c_{1}^{3}}{9}\right )}{4 {\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{{2}/{3}} x^{3} c_{1}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.205 (sec). Leaf size: 400
DSolve[(5*x^3*y[x]^2+2*y[x])+(3*x^4*y[x]+2*x)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {-2 x^2+\frac {2 x^4}{\sqrt [3]{\frac {27 c_1 x^{10}}{2}-x^6+\frac {3}{2} \sqrt {3} \sqrt {c_1 x^{16} \left (-4+27 c_1 x^4\right )}}}+2^{2/3} \sqrt [3]{27 c_1 x^{10}-2 x^6+3 \sqrt {3} \sqrt {c_1 x^{16} \left (-4+27 c_1 x^4\right )}}}{6 x^5} \\
y(x)\to \frac {-4 x^2-\frac {2 \left (1+i \sqrt {3}\right ) x^4}{\sqrt [3]{\frac {27 c_1 x^{10}}{2}-x^6+\frac {3}{2} \sqrt {3} \sqrt {c_1 x^{16} \left (-4+27 c_1 x^4\right )}}}+i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{27 c_1 x^{10}-2 x^6+3 \sqrt {3} \sqrt {c_1 x^{16} \left (-4+27 c_1 x^4\right )}}}{12 x^5} \\
y(x)\to -\frac {4 x^2-\frac {2 i \left (\sqrt {3}+i\right ) x^4}{\sqrt [3]{\frac {27 c_1 x^{10}}{2}-x^6+\frac {3}{2} \sqrt {3} \sqrt {c_1 x^{16} \left (-4+27 c_1 x^4\right )}}}+2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{27 c_1 x^{10}-2 x^6+3 \sqrt {3} \sqrt {c_1 x^{16} \left (-4+27 c_1 x^4\right )}}}{12 x^5} \\
\end{align*}