12.5.35 problem 32
Internal
problem
ID
[1659]
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
:
32
Date
solved
:
Monday, January 27, 2025 at 05:19:21 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} y^{\prime }&=\frac {y}{y-2 x} \end{align*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 302
dsolve(diff(y(x),x)=y(x)/(y(x)-2*x),y(x), singsol=all)
\begin{align*}
y &= \frac {\left (-12 c_1 +8 x^{3}+4 \sqrt {3}\, \sqrt {c_1 \left (-4 x^{3}+3 c_1 \right )}\right )^{{1}/{3}}}{2}+\frac {2 x^{2}}{\left (-12 c_1 +8 x^{3}+4 \sqrt {3}\, \sqrt {c_1 \left (-4 x^{3}+3 c_1 \right )}\right )^{{1}/{3}}}+x \\
y &= \frac {\left (-1-i \sqrt {3}\right ) \left (-12 c_1 +8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{1}/{3}}}{4}+\frac {\left (i x \sqrt {3}-x +\left (-12 c_1 +8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{1}/{3}}\right ) x}{\left (-12 c_1 +8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (-12 c_1 +8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{1}/{3}}}{4}-\frac {x \left (i x \sqrt {3}+x -\left (-12 c_1 +8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{1}/{3}}\right )}{\left (-12 c_1 +8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+3 c_1^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 28.623 (sec). Leaf size: 479
DSolve[D[y[x],x]==y[x]/(y[x]-2*x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2} x^2}{\sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}+x \\
y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}{2 \sqrt [3]{2}}-\frac {\left (1+i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}+x \\
y(x)\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}{2 \sqrt [3]{2}}+\frac {i \left (\sqrt {3}+i\right ) x^2}{2^{2/3} \sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}+x \\
y(x)\to 0 \\
y(x)\to -\frac {i \left (\sqrt [3]{x^3}-x\right ) \left (\left (\sqrt {3}-i\right ) \sqrt [3]{x^3}-2 i x\right )}{2 x} \\
y(x)\to \frac {i \left (\sqrt [3]{x^3}-x\right ) \left (\left (\sqrt {3}+i\right ) \sqrt [3]{x^3}+2 i x\right )}{2 x} \\
y(x)\to \sqrt [3]{x^3}+\frac {\left (x^3\right )^{2/3}}{x}+x \\
\end{align*}