61.22.32 problem 32
Internal
problem
ID
[12278]
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
:
32
Date
solved
:
Friday, March 14, 2025 at 04:41:25 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} y y^{\prime }-y&=\frac {A}{\sqrt {x}} \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 222
ode:=y(x)*diff(y(x),x)-y(x) = A/x^(1/2);
dsolve(ode,y(x), singsol=all);
\[
\frac {-2^{{2}/{3}} \left (-A^{2} x^{{3}/{2}}\right )^{{1}/{3}} \left (-\operatorname {AiryBi}\left (-\frac {2^{{1}/{3}} \left (-A^{2} x^{{3}/{2}}\right )^{{2}/{3}} \left (y-x \right )}{2 A^{2} x}\right ) c_{1} +\operatorname {AiryAi}\left (-\frac {2^{{1}/{3}} \left (-A^{2} x^{{3}/{2}}\right )^{{2}/{3}} \left (y-x \right )}{2 A^{2} x}\right )\right )-2 A \left (-\operatorname {AiryBi}\left (1, -\frac {2^{{1}/{3}} \left (-A^{2} x^{{3}/{2}}\right )^{{2}/{3}} \left (y-x \right )}{2 A^{2} x}\right ) c_{1} +\operatorname {AiryAi}\left (1, -\frac {2^{{1}/{3}} \left (-A^{2} x^{{3}/{2}}\right )^{{2}/{3}} \left (y-x \right )}{2 A^{2} x}\right )\right )}{2^{{2}/{3}} \left (-A^{2} x^{{3}/{2}}\right )^{{1}/{3}} \operatorname {AiryBi}\left (-\frac {2^{{1}/{3}} \left (-A^{2} x^{{3}/{2}}\right )^{{2}/{3}} \left (y-x \right )}{2 A^{2} x}\right )+2 \operatorname {AiryBi}\left (1, -\frac {2^{{1}/{3}} \left (-A^{2} x^{{3}/{2}}\right )^{{2}/{3}} \left (y-x \right )}{2 A^{2} x}\right ) A} = 0
\]
✓ Mathematica. Time used: 0.317 (sec). Leaf size: 139
ode=y[x]*D[y[x],x]-y[x]==A*x^(-1/2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {\sqrt [3]{-1} 2^{2/3} \sqrt {x} \operatorname {AiryAi}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} (x-y(x))}{A^{2/3}}\right )+2 \sqrt [3]{A} \operatorname {AiryAiPrime}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} (x-y(x))}{A^{2/3}}\right )}{\sqrt [3]{-1} 2^{2/3} \sqrt {x} \operatorname {AiryBi}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} (x-y(x))}{A^{2/3}}\right )+2 \sqrt [3]{A} \operatorname {AiryBiPrime}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} (x-y(x))}{A^{2/3}}\right )}+c_1=0,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
A = symbols("A")
y = Function("y")
ode = Eq(-A/sqrt(x) + y(x)*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -A/(sqrt(x)*y(x)) + Derivative(y(x), x) - 1 cannot be solved by the factorable group method