15.5.14 problem 14
Internal
problem
ID
[2950]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
9,
page
38
Problem
number
:
14
Date
solved
:
Monday, January 27, 2025 at 07:00:18 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} \left (2 x +3 x^{2} y\right ) y^{\prime }+y+2 x y^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.011 (sec). Leaf size: 376
dsolve((2*x+3*x^2*y(x))*diff(y(x),x)+(y(x)+2*y(x)^2*x)=0,y(x), singsol=all)
\begin{align*}
y &= \frac {2 c_1^{2} 2^{{1}/{3}}-2 c_1 \left (\left (3 \sqrt {\frac {-12 c_1 +81 x}{x}}\, x -2 c_1 +27 x \right ) c_1^{2}\right )^{{1}/{3}}+2^{{2}/{3}} \left (\left (3 \sqrt {\frac {-12 c_1 +81 x}{x}}\, x -2 c_1 +27 x \right ) c_1^{2}\right )^{{2}/{3}}}{6 c_1 x \left (\left (3 \sqrt {\frac {-12 c_1 +81 x}{x}}\, x -2 c_1 +27 x \right ) c_1^{2}\right )^{{1}/{3}}} \\
y &= \frac {2 \left (i \sqrt {3}-1\right ) c_1^{2} 2^{{1}/{3}}-\left (\left (1+i \sqrt {3}\right ) \left (\left (3 \sqrt {\frac {-12 c_1 +81 x}{x}}\, x -2 c_1 +27 x \right ) c_1^{2}\right )^{{1}/{3}} 2^{{2}/{3}}+4 c_1 \right ) \left (\left (3 \sqrt {\frac {-12 c_1 +81 x}{x}}\, x -2 c_1 +27 x \right ) c_1^{2}\right )^{{1}/{3}}}{12 \left (\left (3 \sqrt {\frac {-12 c_1 +81 x}{x}}\, x -2 c_1 +27 x \right ) c_1^{2}\right )^{{1}/{3}} x c_1} \\
y &= \frac {-2 \left (1+i \sqrt {3}\right ) c_1^{2} 2^{{1}/{3}}+\left (\left (i \sqrt {3}-1\right ) \left (\left (3 \sqrt {\frac {-12 c_1 +81 x}{x}}\, x -2 c_1 +27 x \right ) c_1^{2}\right )^{{1}/{3}} 2^{{2}/{3}}-4 c_1 \right ) \left (\left (3 \sqrt {\frac {-12 c_1 +81 x}{x}}\, x -2 c_1 +27 x \right ) c_1^{2}\right )^{{1}/{3}}}{12 \left (\left (3 \sqrt {\frac {-12 c_1 +81 x}{x}}\, x -2 c_1 +27 x \right ) c_1^{2}\right )^{{1}/{3}} x c_1} \\
\end{align*}
✓ Solution by Mathematica
Time used: 31.110 (sec). Leaf size: 380
DSolve[(2*x+3*x^2*y[x])*D[y[x],x]+(y[x]+2*y[x]^2*x)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {1}{6} \left (\frac {2}{\sqrt [3]{\frac {3}{2} \sqrt {3} \sqrt {c_1 x^7 (-4+27 c_1 x)}+\frac {27 c_1 x^4}{2}-x^3}}+\frac {2^{2/3} \sqrt [3]{3 \sqrt {3} \sqrt {c_1 x^7 (-4+27 c_1 x)}+27 c_1 x^4-2 x^3}}{x^2}-\frac {2}{x}\right ) \\
y(x)\to \frac {1}{12} \left (-\frac {2 \left (1+i \sqrt {3}\right )}{\sqrt [3]{\frac {3}{2} \sqrt {3} \sqrt {c_1 x^7 (-4+27 c_1 x)}+\frac {27 c_1 x^4}{2}-x^3}}+\frac {i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{3 \sqrt {3} \sqrt {c_1 x^7 (-4+27 c_1 x)}+27 c_1 x^4-2 x^3}}{x^2}-\frac {4}{x}\right ) \\
y(x)\to \frac {1}{12} \left (\frac {2 i \left (\sqrt {3}+i\right )}{\sqrt [3]{\frac {3}{2} \sqrt {3} \sqrt {c_1 x^7 (-4+27 c_1 x)}+\frac {27 c_1 x^4}{2}-x^3}}-\frac {2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{3 \sqrt {3} \sqrt {c_1 x^7 (-4+27 c_1 x)}+27 c_1 x^4-2 x^3}}{x^2}-\frac {4}{x}\right ) \\
\end{align*}