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
:
Monday, January 27, 2025 at 05:19:13 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} y^{\prime }&=\frac {x +2 y}{2 x +y} \end{align*}
✓ Solution by Maple
Time used: 0.213 (sec). Leaf size: 267
dsolve(diff(y(x),x)=(x+2*y(x))/(2*x+y(x)),y(x), singsol=all)
\begin{align*}
y &= \frac {x \left (\frac {c_1 \left (\left (27 c_1 x +3 \sqrt {3}\, \sqrt {27 c_1^{2} x^{2}-1}\right )^{{2}/{3}}+3\right )}{3 x \left (27 c_1 x +3 \sqrt {3}\, \sqrt {27 c_1^{2} x^{2}-1}\right )^{{1}/{3}}}+c_1^{2}\right )}{c_1^{2}} \\
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 x c_1 \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 x c_1 \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*}
✓ Solution by Mathematica
Time used: 28.664 (sec). Leaf size: 382
DSolve[D[y[x],x]==(x+2*y[x])/(2*x+y[x]),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*}