79.1.4 problem (d)
Internal
problem
ID
[21081]
Book
:
Ordinary
Differential
Equations.
By
Wolfgang
Walter.
Graduate
texts
in
Mathematics.
Springer.
NY.
QA372.W224
1998
Section
:
Chapter
1.
First
order
equations:
Some
integrable
cases.
Excercises
XII
at
page
23
Problem
number
:
(d)
Date
solved
:
Thursday, October 02, 2025 at 07:06:37 PM
CAS
classification
:
[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} y^{\prime }&=\frac {x +2 y+1}{2 x +2+y} \end{align*}
✓ Maple. Time used: 0.220 (sec). Leaf size: 116
ode:=diff(y(x),x) = (x+2*y(x)+1)/(2*x+y(x)+2);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\frac {1}{2}+\frac {\left (1-i \sqrt {3}\right ) \left (3 \sqrt {3}\, \sqrt {27 c_1^{2} \left (x +1\right )^{2}-1}-27 \left (x +1\right ) c_1 \right )^{{2}/{3}}}{6}+\frac {i \sqrt {3}}{2}+\left (3 \sqrt {3}\, \sqrt {27 c_1^{2} \left (x +1\right )^{2}-1}-27 c_1 x -27 c_1 \right )^{{1}/{3}} \left (x +1\right ) c_1}{\left (3 \sqrt {3}\, \sqrt {27 c_1^{2} \left (x +1\right )^{2}-1}-27 \left (x +1\right ) c_1 \right )^{{1}/{3}} c_1}
\]
✓ Mathematica. Time used: 60.123 (sec). Leaf size: 1598
ode=D[y[x],x]==(x+2*y[x]+1)/(2*x+y[x]+2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Too large to display
✓ Sympy. Time used: 98.379 (sec). Leaf size: 371
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((-x - 2*y(x) - 1)/(2*x + y(x) + 2) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\frac {2 \cdot 3^{\frac {2}{3}} i C_{1}}{3 \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2} + 54 x + 27} + 9\right )}} + \sqrt {3} x - i x + \frac {3^{\frac {5}{6}} \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2} + 54 x + 27} + 9\right )}}{3} + \frac {\sqrt [3]{3} i \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2} + 54 x + 27} + 9\right )}}{3} + \sqrt {3} - i}{\sqrt {3} - i}, \ y{\left (x \right )} = \frac {- \frac {2 \cdot 3^{\frac {2}{3}} i C_{1}}{3 \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2} + 54 x + 27} + 9\right )}} + \sqrt {3} x + i x + \frac {3^{\frac {5}{6}} \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2} + 54 x + 27} + 9\right )}}{3} - \frac {\sqrt [3]{3} i \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2} + 54 x + 27} + 9\right )}}{3} + \sqrt {3} + i}{\sqrt {3} + i}, \ y{\left (x \right )} = \frac {3^{\frac {2}{3}} C_{1}}{3 \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2} + 54 x + 27} + 9\right )}} + x - \frac {\sqrt [3]{3} \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2} + 54 x + 27} + 9\right )}}{3} + 1\right ]
\]