12.5.34 problem 31
Internal
problem
ID
[1658]
Book
:
Elementary
differential
equations
with
boundary
value
problems.
William
F.
Trench.
Brooks/Cole
2001
Section
:
Chapter
2,
First
order
equations.
Transformation
of
Nonlinear
Equations
into
Separable
Equations.
Section
2.4
Page
68
Problem
number
:
31
Date
solved
:
Saturday, March 29, 2025 at 11:14:33 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} y^{\prime }&=\frac {x +2 y}{2 x +y} \end{align*}
✓ Maple. Time used: 0.027 (sec). Leaf size: 259
ode:=diff(y(x),x) = (x+2*y(x))/(2*x+y(x));
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (27 c_1 x +3 \sqrt {3}\, \sqrt {27 c_1^{2} x^{2}-1}\right )^{{2}/{3}}+3}{3 c_1 \left (27 c_1 x +3 \sqrt {3}\, \sqrt {27 c_1^{2} x^{2}-1}\right )^{{1}/{3}}}+x \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (27 c_1 x +3 \sqrt {3}\, \sqrt {27 c_1^{2} x^{2}-1}\right )^{{2}/{3}}-6 c_1 x \left (27 c_1 x +3 \sqrt {3}\, \sqrt {27 c_1^{2} x^{2}-1}\right )^{{1}/{3}}-3 i \sqrt {3}+3}{6 \left (27 c_1 x +3 \sqrt {3}\, \sqrt {27 c_1^{2} x^{2}-1}\right )^{{1}/{3}} c_1} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (27 c_1 x +3 \sqrt {3}\, \sqrt {27 c_1^{2} x^{2}-1}\right )^{{2}/{3}}+6 c_1 x \left (27 c_1 x +3 \sqrt {3}\, \sqrt {27 c_1^{2} x^{2}-1}\right )^{{1}/{3}}-3 i \sqrt {3}-3}{6 \left (27 c_1 x +3 \sqrt {3}\, \sqrt {27 c_1^{2} x^{2}-1}\right )^{{1}/{3}} c_1} \\
\end{align*}
✓ Mathematica. Time used: 28.664 (sec). Leaf size: 382
ode=D[y[x],x]==(x+2*y[x])/(2*x+y[x]);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{\sqrt {3} \sqrt {27 e^{4 c_1} x^2+e^{6 c_1}}-9 e^{2 c_1} x}}{3^{2/3}}-\frac {e^{2 c_1}}{\sqrt [3]{3} \sqrt [3]{\sqrt {3} \sqrt {27 e^{4 c_1} x^2+e^{6 c_1}}-9 e^{2 c_1} x}}+x \\
y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {3} \sqrt {27 e^{4 c_1} x^2+e^{6 c_1}}-9 e^{2 c_1} x}}{2\ 3^{2/3}}+\frac {\left (1+i \sqrt {3}\right ) e^{2 c_1}}{2 \sqrt [3]{3} \sqrt [3]{\sqrt {3} \sqrt {27 e^{4 c_1} x^2+e^{6 c_1}}-9 e^{2 c_1} x}}+x \\
y(x)\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {3} \sqrt {27 e^{4 c_1} x^2+e^{6 c_1}}-9 e^{2 c_1} x}}{2\ 3^{2/3}}+\frac {\left (1-i \sqrt {3}\right ) e^{2 c_1}}{2 \sqrt [3]{3} \sqrt [3]{\sqrt {3} \sqrt {27 e^{4 c_1} x^2+e^{6 c_1}}-9 e^{2 c_1} x}}+x \\
\end{align*}
✓ Sympy. Time used: 160.456 (sec). Leaf size: 301
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((-x - 2*y(x))/(2*x + y(x)) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\frac {2 \cdot 3^{\frac {2}{3}} i C_{1}}{3 \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2}}\right )}} + \sqrt {3} x - i x + \frac {3^{\frac {5}{6}} \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2}}\right )}}{3} + \frac {\sqrt [3]{3} i \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2}}\right )}}{3}}{\sqrt {3} - i}, \ y{\left (x \right )} = \frac {- \frac {2 \cdot 3^{\frac {2}{3}} i C_{1}}{3 \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2}}\right )}} + \sqrt {3} x + i x + \frac {3^{\frac {5}{6}} \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2}}\right )}}{3} - \frac {\sqrt [3]{3} i \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2}}\right )}}{3}}{\sqrt {3} + i}, \ y{\left (x \right )} = \frac {3^{\frac {2}{3}} C_{1}}{3 \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2}}\right )}} + x - \frac {\sqrt [3]{3} \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2}}\right )}}{3}\right ]
\]