89.8.15 problem 17
Internal
problem
ID
[24475]
Book
:
A
short
course
in
Differential
Equations.
Earl
D.
Rainville.
Second
edition.
1958.
Macmillan
Publisher,
NY.
CAT
58-5010
Section
:
Chapter
4.
Additional
topics
on
equations
of
first
order
and
first
degree.
Exercises
at
page
66
Problem
number
:
17
Date
solved
:
Thursday, October 02, 2025 at 10:40:43 PM
CAS
classification
:
[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} x +2 y-1-\left (2 x +y-5\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.120 (sec). Leaf size: 116
ode:=x+2*y(x)-1-(2*x+y(x)-5)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\frac {1}{2}+\frac {\left (1-i \sqrt {3}\right ) \left (3 \sqrt {3}\, \sqrt {27 c_1^{2} \left (x -3\right )^{2}-1}-27 c_1 \left (x -3\right )\right )^{{2}/{3}}}{6}+\frac {i \sqrt {3}}{2}+\left (3 \sqrt {3}\, \sqrt {27 c_1^{2} \left (x -3\right )^{2}-1}-27 c_1 x +81 c_1 \right )^{{1}/{3}} \left (x -4\right ) c_1}{\left (3 \sqrt {3}\, \sqrt {27 c_1^{2} \left (x -3\right )^{2}-1}-27 c_1 \left (x -3\right )\right )^{{1}/{3}} c_1}
\]
✓ Mathematica. Time used: 60.108 (sec). Leaf size: 1626
ode=( x+2*y[x] -1)-( 2*x+y[x]-5 )*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Too large to display
✓ Sympy. Time used: 97.786 (sec). Leaf size: 377
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x - (2*x + y(x) - 5)*Derivative(y(x), x) + 2*y(x) - 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\frac {2 \cdot 3^{\frac {2}{3}} i C_{1}}{3 \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2} - 162 x + 243} - 27\right )}} + \sqrt {3} x - i x + \frac {3^{\frac {5}{6}} \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2} - 162 x + 243} - 27\right )}}{3} + \frac {\sqrt [3]{3} i \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2} - 162 x + 243} - 27\right )}}{3} - 4 \sqrt {3} + 4 i}{\sqrt {3} - i}, \ y{\left (x \right )} = \frac {- \frac {2 \cdot 3^{\frac {2}{3}} i C_{1}}{3 \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2} - 162 x + 243} - 27\right )}} + \sqrt {3} x + i x + \frac {3^{\frac {5}{6}} \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2} - 162 x + 243} - 27\right )}}{3} - \frac {\sqrt [3]{3} i \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2} - 162 x + 243} - 27\right )}}{3} - 4 \sqrt {3} - 4 i}{\sqrt {3} + i}, \ y{\left (x \right )} = \frac {3^{\frac {2}{3}} C_{1}}{3 \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2} - 162 x + 243} - 27\right )}} + x - \frac {\sqrt [3]{3} \sqrt [3]{C_{1} \left (9 x + \sqrt {3} \sqrt {C_{1} + 27 x^{2} - 162 x + 243} - 27\right )}}{3} - 4\right ]
\]