41.2.42 problem 40
Internal
problem
ID
[8744]
Book
:
Ordinary
differential
equations
and
calculus
of
variations.
Makarets
and
Reshetnyak.
Wold
Scientific.
Singapore.
1995
Section
:
Chapter
1.
First
order
differential
equations.
Section
1.2
Homogeneous
equations
problems.
page
12
Problem
number
:
40
Date
solved
:
Tuesday, September 30, 2025 at 05:48:25 PM
CAS
classification
:
[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} y^{\prime }&=\frac {x -2 y+5}{y-2 x -4} \end{align*}
✓ Maple. Time used: 0.235 (sec). Leaf size: 115
ode:=diff(y(x),x) = (x-2*y(x)+5)/(y(x)-2*x-4);
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {\left (i \sqrt {3}-1\right ) \left (3 \sqrt {3}\, \sqrt {27 c_1^{2} \left (x +1\right )^{2}-1}+27 \left (x +1\right ) c_1 \right )^{{2}/{3}}-3 i \sqrt {3}-3+6 \left (3 \sqrt {3}\, \sqrt {27 c_1^{2} \left (x +1\right )^{2}-1}+27 c_1 x +27 c_1 \right )^{{1}/{3}} \left (x -1\right ) c_1}{6 \left (3 \sqrt {3}\, \sqrt {27 c_1^{2} \left (x +1\right )^{2}-1}+27 \left (x +1\right ) c_1 \right )^{{1}/{3}} c_1}
\]
✓ Mathematica. Time used: 60.117 (sec). Leaf size: 1601
ode=D[y[x],x]==(x-2*y[x]+5)/(y[x]-2*x-4);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Too large to display
✓ Sympy. Time used: 102.790 (sec). Leaf size: 371
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(Derivative(y(x), x) - (x - 2*y(x) + 5)/(-2*x + y(x) - 4),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {- \frac {2 \cdot 3^{\frac {2}{3}} i C_{1}}{3 \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {- C_{1} + 27 x^{2} + 54 x + 27} + 9\right )}} - \sqrt {3} x + i x + \frac {3^{\frac {5}{6}} \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {- C_{1} + 27 x^{2} + 54 x + 27} + 9\right )}}{3} + \frac {\sqrt [3]{3} i \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {- C_{1} + 27 x^{2} + 54 x + 27} + 9\right )}}{3} + \sqrt {3} - i}{\sqrt {3} - i}, \ y{\left (x \right )} = \frac {\frac {2 \cdot 3^{\frac {2}{3}} i C_{1}}{3 \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {- C_{1} + 27 x^{2} + 54 x + 27} + 9\right )}} - \sqrt {3} x - i x + \frac {3^{\frac {5}{6}} \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {- C_{1} + 27 x^{2} + 54 x + 27} + 9\right )}}{3} - \frac {\sqrt [3]{3} i \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {- C_{1} + 27 x^{2} + 54 x + 27} + 9\right )}}{3} + \sqrt {3} + i}{\sqrt {3} + i}, \ y{\left (x \right )} = - \frac {3^{\frac {2}{3}} C_{1}}{3 \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {- C_{1} + 27 x^{2} + 54 x + 27} + 9\right )}} - x - \frac {\sqrt [3]{3} \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {- C_{1} + 27 x^{2} + 54 x + 27} + 9\right )}}{3} + 1\right ]
\]