75.7.23 problem 198
Internal
problem
ID
[16737]
Book
:
A
book
of
problems
in
ordinary
differential
equations.
M.L.
KRASNOV,
A.L.
KISELYOV,
G.I.
MARKARENKO.
MIR,
MOSCOW.
1983
Section
:
Section
7,
Total
differential
equations.
The
integrating
factor.
Exercises
page
61
Problem
number
:
198
Date
solved
:
Thursday, March 13, 2025 at 08:39:48 AM
CAS
classification
:
[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class A`]]
\begin{align*} x -x y+\left (y+x^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 55
ode:=x-x*y(x)+(y(x)+x^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {2 c_{1} +1-\sqrt {2 c_{1} x^{2}+2 c_{1} +1}}{2 c_{1}} \\
y &= \frac {2 c_{1} +1+\sqrt {2 c_{1} x^{2}+2 c_{1} +1}}{2 c_{1}} \\
\end{align*}
✓ Mathematica. Time used: 5.282 (sec). Leaf size: 295
ode=( x -x*y[x] )+( y[x]+x^2)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -x^2+\frac {1}{\frac {1}{x^2+1}-\frac {1+i}{\left (x^2+1\right ) \sqrt {-2 \left (x^2+1\right ) \cosh \left (\frac {2 c_1}{9}\right )-2 \left (x^2+1\right ) \sinh \left (\frac {2 c_1}{9}\right )+2 i}}} \\
y(x)\to -x^2+\frac {1}{\frac {1}{x^2+1}+\frac {1+i}{\left (x^2+1\right ) \sqrt {-2 \left (x^2+1\right ) \cosh \left (\frac {2 c_1}{9}\right )-2 \left (x^2+1\right ) \sinh \left (\frac {2 c_1}{9}\right )+2 i}}} \\
y(x)\to -x^2+\frac {1}{\frac {1}{x^2+1}-\frac {1+i}{\sqrt {2} \left (x^2+1\right ) \sqrt {\left (x^2+1\right ) \cosh \left (\frac {2 c_1}{9}\right )+\left (x^2+1\right ) \sinh \left (\frac {2 c_1}{9}\right )+i}}} \\
y(x)\to -x^2+\frac {1}{\frac {1}{x^2+1}+\frac {1+i}{\sqrt {2} \left (x^2+1\right ) \sqrt {\left (x^2+1\right ) \cosh \left (\frac {2 c_1}{9}\right )+\left (x^2+1\right ) \sinh \left (\frac {2 c_1}{9}\right )+i}}} \\
y(x)\to 1 \\
y(x)\to \frac {1}{2} \left (1-x^2\right ) \\
\end{align*}
✓ Sympy. Time used: 1.255 (sec). Leaf size: 51
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x*y(x) + x + (x**2 + y(x))*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {2 C_{1} - \sqrt {- 2 C_{1} x^{2} - 2 C_{1} + 1} - 1}{2 C_{1}}, \ y{\left (x \right )} = \frac {2 C_{1} + \sqrt {- 2 C_{1} x^{2} - 2 C_{1} + 1} - 1}{2 C_{1}}\right ]
\]