61.24.16 problem 16
Internal
problem
ID
[12350]
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
1.3.3-2.
Problem
number
:
16
Date
solved
:
Friday, March 14, 2025 at 04:44:17 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} y y^{\prime }-a \left (1-\frac {b}{\sqrt {x}}\right ) y&=\frac {a^{2} b}{\sqrt {x}} \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 269
ode:=y(x)*diff(y(x),x)-a*(1-b/x^(1/2))*y(x) = a^2*b/x^(1/2);
dsolve(ode,y(x), singsol=all);
\[
\frac {\left (b^{2}\right )^{{1}/{3}} c_{1} 2^{{2}/{3}} \left (-\sqrt {x}+b \right ) \operatorname {AiryBi}\left (\frac {\left (-2 b a \sqrt {x}+\left (b^{2}+x \right ) a -y\right ) 2^{{1}/{3}}}{2 \left (b^{2}\right )^{{1}/{3}} a}\right )+2 \operatorname {AiryBi}\left (1, \frac {\left (-2 b a \sqrt {x}+\left (b^{2}+x \right ) a -y\right ) 2^{{1}/{3}}}{2 \left (b^{2}\right )^{{1}/{3}} a}\right ) c_{1} b -2 \operatorname {AiryAi}\left (1, \frac {\left (-2 b a \sqrt {x}+\left (b^{2}+x \right ) a -y\right ) 2^{{1}/{3}}}{2 \left (b^{2}\right )^{{1}/{3}} a}\right ) b -\left (b^{2}\right )^{{1}/{3}} 2^{{2}/{3}} \left (-\sqrt {x}+b \right ) \operatorname {AiryAi}\left (\frac {\left (-2 b a \sqrt {x}+\left (b^{2}+x \right ) a -y\right ) 2^{{1}/{3}}}{2 \left (b^{2}\right )^{{1}/{3}} a}\right )}{\left (b^{2}\right )^{{1}/{3}} 2^{{2}/{3}} \left (-\sqrt {x}+b \right ) \operatorname {AiryBi}\left (\frac {\left (-2 b a \sqrt {x}+\left (b^{2}+x \right ) a -y\right ) 2^{{1}/{3}}}{2 \left (b^{2}\right )^{{1}/{3}} a}\right )+2 \operatorname {AiryBi}\left (1, \frac {\left (-2 b a \sqrt {x}+\left (b^{2}+x \right ) a -y\right ) 2^{{1}/{3}}}{2 \left (b^{2}\right )^{{1}/{3}} a}\right ) b} = 0
\]
✓ Mathematica. Time used: 1.799 (sec). Leaf size: 323
ode=y[x]*D[y[x],x]-a*(1-b*x^(-1/2))*y[x]==a^2*b*x^(-1/2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {\sqrt [3]{-1} 2^{2/3} \sqrt [3]{\left (b-\sqrt {x}\right )^3} \operatorname {AiryAi}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} \left (\left (b-\sqrt {x}\right )^3\right )^{2/3} \left (a \left (b-\sqrt {x}\right )^2-y(x)\right )}{a b^{2/3} \left (b-\sqrt {x}\right )^2}\right )-2 \sqrt [3]{b} \operatorname {AiryAiPrime}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} \left (\left (b-\sqrt {x}\right )^3\right )^{2/3} \left (a \left (b-\sqrt {x}\right )^2-y(x)\right )}{a b^{2/3} \left (b-\sqrt {x}\right )^2}\right )}{\sqrt [3]{-1} 2^{2/3} \sqrt [3]{\left (b-\sqrt {x}\right )^3} \operatorname {AiryBi}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} \left (\left (b-\sqrt {x}\right )^3\right )^{2/3} \left (a \left (b-\sqrt {x}\right )^2-y(x)\right )}{a b^{2/3} \left (b-\sqrt {x}\right )^2}\right )-2 \sqrt [3]{b} \operatorname {AiryBiPrime}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} \left (\left (b-\sqrt {x}\right )^3\right )^{2/3} \left (a \left (b-\sqrt {x}\right )^2-y(x)\right )}{a b^{2/3} \left (b-\sqrt {x}\right )^2}\right )}+c_1=0,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(-a**2*b/sqrt(x) - a*(-b/sqrt(x) + 1)*y(x) + y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -a**2*b/(sqrt(x)*y(x)) + a*b/sqrt(x) - a + Derivative(y(x), x) cannot be solved by the factorable group method