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
:
Sunday, March 30, 2025 at 01:01:01 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 y^{2} x&=0 \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 376
ode:=(2*x+3*x^2*y(x))*diff(y(x),x)+y(x)+2*x*y(x)^2 = 0;
dsolve(ode,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 c_1^{2} \left (i \sqrt {3}-1\right ) 2^{{1}/{3}}-\left (\left (3 \sqrt {\frac {-12 c_1 +81 x}{x}}\, x -2 c_1 +27 x \right ) c_1^{2}\right )^{{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 )}{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}} c_1 x} \\
y &= \frac {-2 c_1^{2} \left (1+i \sqrt {3}\right ) 2^{{1}/{3}}+\left (\left (3 \sqrt {\frac {-12 c_1 +81 x}{x}}\, x -2 c_1 +27 x \right ) c_1^{2}\right )^{{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 )}{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}} c_1 x} \\
\end{align*}
✓ Mathematica. Time used: 31.11 (sec). Leaf size: 380
ode=(2*x+3*x^2*y[x])*D[y[x],x]+(y[x]+2*y[x]^2*x)==0;
ic={};
DSolve[{ode,ic},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*}
✓ Sympy. Time used: 42.798 (sec). Leaf size: 430
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x*y(x)**2 + (3*x**2*y(x) + 2*x)*Derivative(y(x), x) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {- 2^{\frac {2}{3}} \sqrt [3]{- \frac {27 C_{1}}{x^{2}} + 3 \sqrt {3} \sqrt {\frac {C_{1} \left (27 C_{1} - \frac {4}{x}\right )}{x^{4}}} + \frac {2}{x^{3}}} - 2^{\frac {2}{3}} \sqrt {3} i \sqrt [3]{- \frac {27 C_{1}}{x^{2}} + 3 \sqrt {3} \sqrt {\frac {C_{1} \left (27 C_{1} - \frac {4}{x}\right )}{x^{4}}} + \frac {2}{x^{3}}} - \frac {2}{x} + \frac {2 \sqrt {3} i}{x} + \frac {4 \sqrt [3]{2}}{x^{2} \sqrt [3]{- \frac {27 C_{1}}{x^{2}} + 3 \sqrt {3} \sqrt {\frac {C_{1} \left (27 C_{1} - \frac {4}{x}\right )}{x^{4}}} + \frac {2}{x^{3}}}}}{6 \left (1 - \sqrt {3} i\right )}, \ y{\left (x \right )} = \frac {- 2^{\frac {2}{3}} \sqrt [3]{- \frac {27 C_{1}}{x^{2}} + 3 \sqrt {3} \sqrt {\frac {C_{1} \left (27 C_{1} - \frac {4}{x}\right )}{x^{4}}} + \frac {2}{x^{3}}} + 2^{\frac {2}{3}} \sqrt {3} i \sqrt [3]{- \frac {27 C_{1}}{x^{2}} + 3 \sqrt {3} \sqrt {\frac {C_{1} \left (27 C_{1} - \frac {4}{x}\right )}{x^{4}}} + \frac {2}{x^{3}}} - \frac {2}{x} - \frac {2 \sqrt {3} i}{x} + \frac {4 \sqrt [3]{2}}{x^{2} \sqrt [3]{- \frac {27 C_{1}}{x^{2}} + 3 \sqrt {3} \sqrt {\frac {C_{1} \left (27 C_{1} - \frac {4}{x}\right )}{x^{4}}} + \frac {2}{x^{3}}}}}{6 \left (1 + \sqrt {3} i\right )}, \ y{\left (x \right )} = - \frac {2^{\frac {2}{3}} \sqrt [3]{- \frac {27 C_{1}}{x^{2}} + 3 \sqrt {3} \sqrt {\frac {C_{1} \left (27 C_{1} - \frac {4}{x}\right )}{x^{4}}} + \frac {2}{x^{3}}}}{6} - \frac {1}{3 x} - \frac {\sqrt [3]{2}}{3 x^{2} \sqrt [3]{- \frac {27 C_{1}}{x^{2}} + 3 \sqrt {3} \sqrt {\frac {C_{1} \left (27 C_{1} - \frac {4}{x}\right )}{x^{4}}} + \frac {2}{x^{3}}}}\right ]
\]