9.3.2 problem 2
Internal
problem
ID
[2895]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
7,
page
28
Problem
number
:
2
Date
solved
:
Tuesday, September 30, 2025 at 06:01:12 AM
CAS
classification
:
[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class C`], _dAlembert]
\begin{align*} x +\left (x -2 y+2\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.236 (sec). Leaf size: 219
ode:=x+(x-2*y(x)+2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (4+\left (-12 x^{3}-12 x^{2}\right ) c_1^{3}\right ) \left (-c_1^{3} x^{3}+\sqrt {-2 c_1^{3} x^{3}+1}+1\right )^{{2}/{3}}-6 \left (\left (c_1^{3} x^{3}-\frac {\sqrt {-2 c_1^{3} x^{3}+1}}{3}-\frac {1}{3}\right ) \left (1+i \sqrt {3}\right ) \left (-c_1^{3} x^{3}+\sqrt {-2 c_1^{3} x^{3}+1}+1\right )^{{1}/{3}}+\left (c_1^{3} x^{3}-\frac {2 \sqrt {-2 c_1^{3} x^{3}+1}}{3}-\frac {2}{3}\right ) x \left (i \sqrt {3}-1\right ) c_1 \right ) \left (x +1\right ) c_1}{{\left (\left (\sqrt {3}+i\right ) \left (-c_1^{3} x^{3}+\sqrt {-2 c_1^{3} x^{3}+1}+1\right )^{{2}/{3}}+\left (2 i \left (-c_1^{3} x^{3}+\sqrt {-2 c_1^{3} x^{3}+1}+1\right )^{{1}/{3}}+\left (i-\sqrt {3}\right ) x c_1 \right ) x c_1 \right )}^{2} c_1}
\]
✓ Mathematica. Time used: 60.023 (sec). Leaf size: 445
ode=x+(x-2*y[x]+2)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {x+2}{2}-\frac {1}{2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^6+16 e^{12 c_1}\right )-24 \text {$\#$1}^4 x^4+8 \text {$\#$1}^3 x^3+9 \text {$\#$1}^2 x^2-6 \text {$\#$1} x+1\&,1\right ]}\\ y(x)&\to \frac {x+2}{2}-\frac {1}{2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^6+16 e^{12 c_1}\right )-24 \text {$\#$1}^4 x^4+8 \text {$\#$1}^3 x^3+9 \text {$\#$1}^2 x^2-6 \text {$\#$1} x+1\&,2\right ]}\\ y(x)&\to \frac {x+2}{2}-\frac {1}{2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^6+16 e^{12 c_1}\right )-24 \text {$\#$1}^4 x^4+8 \text {$\#$1}^3 x^3+9 \text {$\#$1}^2 x^2-6 \text {$\#$1} x+1\&,3\right ]}\\ y(x)&\to \frac {x+2}{2}-\frac {1}{2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^6+16 e^{12 c_1}\right )-24 \text {$\#$1}^4 x^4+8 \text {$\#$1}^3 x^3+9 \text {$\#$1}^2 x^2-6 \text {$\#$1} x+1\&,4\right ]}\\ y(x)&\to \frac {x+2}{2}-\frac {1}{2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^6+16 e^{12 c_1}\right )-24 \text {$\#$1}^4 x^4+8 \text {$\#$1}^3 x^3+9 \text {$\#$1}^2 x^2-6 \text {$\#$1} x+1\&,5\right ]}\\ y(x)&\to \frac {x+2}{2}-\frac {1}{2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^6+16 e^{12 c_1}\right )-24 \text {$\#$1}^4 x^4+8 \text {$\#$1}^3 x^3+9 \text {$\#$1}^2 x^2-6 \text {$\#$1} x+1\&,6\right ]} \end{align*}
✓ Sympy. Time used: 16.399 (sec). Leaf size: 238
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x + (x - 2*y(x) + 2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\frac {x^{2}}{\sqrt [3]{C_{1} - x^{3} + \sqrt {C_{1} \left (C_{1} - 2 x^{3}\right )}}} - \frac {\sqrt [3]{C_{1} - x^{3} + \sqrt {C_{1} \left (C_{1} - 2 x^{3}\right )}}}{2} - \frac {\sqrt {3} i \sqrt [3]{C_{1} - x^{3} + \sqrt {C_{1} \left (C_{1} - 2 x^{3}\right )}}}{2} + 1 - \sqrt {3} i}{1 - \sqrt {3} i}, \ y{\left (x \right )} = \frac {\frac {x^{2}}{\sqrt [3]{C_{1} - x^{3} + \sqrt {C_{1} \left (C_{1} - 2 x^{3}\right )}}} - \frac {\sqrt [3]{C_{1} - x^{3} + \sqrt {C_{1} \left (C_{1} - 2 x^{3}\right )}}}{2} + \frac {\sqrt {3} i \sqrt [3]{C_{1} - x^{3} + \sqrt {C_{1} \left (C_{1} - 2 x^{3}\right )}}}{2} + 1 + \sqrt {3} i}{1 + \sqrt {3} i}, \ y{\left (x \right )} = - \frac {x^{2}}{2 \sqrt [3]{C_{1} - x^{3} + \sqrt {C_{1} \left (C_{1} - 2 x^{3}\right )}}} - \frac {\sqrt [3]{C_{1} - x^{3} + \sqrt {C_{1} \left (C_{1} - 2 x^{3}\right )}}}{2} + 1\right ]
\]