61.22.35 problem 35
Internal
problem
ID
[12281]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.3.
Abel
Equations
of
the
Second
Kind.
subsection
1.3.1-2.
Solvable
equations
and
their
solutions
Problem
number
:
35
Date
solved
:
Friday, March 14, 2025 at 04:41:47 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} y y^{\prime }-y&=A \left (n +2\right ) \left (\sqrt {x}+2 \left (n +2\right ) A +\frac {\left (2 n +3\right ) A^{2}}{\sqrt {x}}\right ) \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 474
ode:=y(x)*diff(y(x),x)-y(x) = A*(n+2)*(x^(1/2)+2*(n+2)*A+(2*n+3)*A^2/x^(1/2));
dsolve(ode,y(x), singsol=all);
\[
\frac {\left (n +2\right ) A \left (\operatorname {BesselI}\left (1+\sqrt {\frac {\left (n +1\right )^{2}}{\left (n +2\right )^{2}}}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y}{\left (n +2\right )^{2} A^{2}}}\right ) c_{1} +\operatorname {BesselK}\left (1+\sqrt {\frac {\left (n +1\right )^{2}}{\left (n +2\right )^{2}}}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y}{\left (n +2\right )^{2} A^{2}}}\right )\right ) \sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y}{\left (n +2\right )^{2} A^{2}}}-\left (A \sqrt {\frac {\left (n +1\right )^{2}}{\left (n +2\right )^{2}}}\, \left (n +2\right )-\sqrt {x}+\left (-n -2\right ) A \right ) \left (c_{1} \operatorname {BesselI}\left (\sqrt {\frac {\left (n +1\right )^{2}}{\left (n +2\right )^{2}}}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y}{\left (n +2\right )^{2} A^{2}}}\right )-\operatorname {BesselK}\left (\sqrt {\frac {\left (n +1\right )^{2}}{\left (n +2\right )^{2}}}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y}{\left (n +2\right )^{2} A^{2}}}\right )\right )}{A \sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y}{\left (n +2\right )^{2} A^{2}}}\, \left (n +2\right ) \operatorname {BesselI}\left (1+\sqrt {\frac {\left (n +1\right )^{2}}{\left (n +2\right )^{2}}}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y}{\left (n +2\right )^{2} A^{2}}}\right )-\operatorname {BesselI}\left (\sqrt {\frac {\left (n +1\right )^{2}}{\left (n +2\right )^{2}}}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y}{\left (n +2\right )^{2} A^{2}}}\right ) \left (A \sqrt {\frac {\left (n +1\right )^{2}}{\left (n +2\right )^{2}}}\, \left (n +2\right )-\sqrt {x}+\left (-n -2\right ) A \right )} = 0
\]
✗ Mathematica
ode=y[x]*D[y[x],x]-y[x]==A*(n+2)*(x^(1/2)+2*(n+2)*A+(2*n+3)*A^2*x^(-1/2));
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
✗ Sympy
from sympy import *
x = symbols("x")
A = symbols("A")
n = symbols("n")
y = Function("y")
ode = Eq(-A*(n + 2)*(A**2*(2*n + 3)/sqrt(x) + A*(2*n + 4) + sqrt(x)) + y(x)*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out