77.1.32 problem 49 (page 56)
Internal
problem
ID
[17851]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
49
(page
56)
Date
solved
:
Monday, March 31, 2025 at 04:36:16 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} \left (x \left (x +y\right )+a^{2}\right ) y^{\prime }&=y \left (x +y\right )+b^{2} \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 115
ode:=(x*(x+y(x))+a^2)*diff(y(x),x) = y(x)*(x+y(x))+b^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {c_1 \,a^{2} b^{2} x +x +\sqrt {\left (a^{2}+b^{2}\right ) \left (-1+\left (a^{4}+a^{2} x^{2}+b^{2} x^{2}\right ) c_1 \right )}}{c_1 \,a^{4}-1} \\
y &= \frac {c_1 \,a^{2} b^{2} x +x -\sqrt {\left (a^{2}+b^{2}\right ) \left (-1+\left (a^{4}+a^{2} x^{2}+b^{2} x^{2}\right ) c_1 \right )}}{c_1 \,a^{4}-1} \\
\end{align*}
✓ Mathematica. Time used: 5.936 (sec). Leaf size: 228
ode=(x*(x+y[x])+a^2)*D[y[x],x]==y[x]*(x+y[x])+b^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {a^2-\frac {1}{\frac {a^2}{a^4+a^2 x^2+b^2 x^2}-\frac {x}{\left (a^4+a^2 x^2+b^2 x^2\right )^{3/2} \sqrt {-\frac {1}{\left (a^2+b^2\right ) \left (a^4+a^2 x^2+b^2 x^2\right )}+c_1}}}+x^2}{x} \\
y(x)\to -\frac {a^2-\frac {1}{\frac {a^2}{a^4+a^2 x^2+b^2 x^2}+\frac {x}{\left (a^4+a^2 x^2+b^2 x^2\right )^{3/2} \sqrt {-\frac {1}{\left (a^2+b^2\right ) \left (a^4+a^2 x^2+b^2 x^2\right )}+c_1}}}+x^2}{x} \\
y(x)\to \frac {b^2 x}{a^2} \\
\end{align*}
✓ Sympy. Time used: 82.125 (sec). Leaf size: 54
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(-b**2 + (a**2 + x*(x + y(x)))*Derivative(y(x), x) - (x + y(x))*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
C_{1} + \frac {a^{2} \log {\left (- a^{2} y{\left (x \right )} + b^{2} x \right )}}{a^{2} + b^{2}} - \frac {a^{2} \log {\left (a^{2} + b^{2} + \left (x + y{\left (x \right )}\right )^{2} \right )}}{2 \left (a^{2} + b^{2}\right )} = 0
\]