2.24.11 Problem 16
Internal
problem
ID
[13575]
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, December 19, 2025 at 07:17:35 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*}
Unknown ode type.
2.24.11.1 ✓ 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);
\[
\frac {\left (b^{2}\right )^{{1}/{3}} c_1 2^{{2}/{3}} \left (-\sqrt {x}+b \right ) \operatorname {AiryBi}\left (\frac {\left (-2 \sqrt {x}\, a b +\left (b^{2}+x \right ) a -y\right ) 2^{{1}/{3}}}{2 \left (b^{2}\right )^{{1}/{3}} a}\right )+2 \operatorname {AiryBi}\left (1, \frac {\left (-2 \sqrt {x}\, a b +\left (b^{2}+x \right ) a -y\right ) 2^{{1}/{3}}}{2 \left (b^{2}\right )^{{1}/{3}} a}\right ) c_1 b -2 \operatorname {AiryAi}\left (1, \frac {\left (-2 \sqrt {x}\, a b +\left (b^{2}+x \right ) a -y\right ) 2^{{1}/{3}}}{2 \left (b^{2}\right )^{{1}/{3}} a}\right ) b -2^{{2}/{3}} \left (b^{2}\right )^{{1}/{3}} \left (-\sqrt {x}+b \right ) \operatorname {AiryAi}\left (\frac {\left (-2 \sqrt {x}\, a b +\left (b^{2}+x \right ) a -y\right ) 2^{{1}/{3}}}{2 \left (b^{2}\right )^{{1}/{3}} a}\right )}{2^{{2}/{3}} \left (b^{2}\right )^{{1}/{3}} \left (-\sqrt {x}+b \right ) \operatorname {AiryBi}\left (\frac {\left (-2 \sqrt {x}\, a b +\left (b^{2}+x \right ) a -y\right ) 2^{{1}/{3}}}{2 \left (b^{2}\right )^{{1}/{3}} a}\right )+2 \operatorname {AiryBi}\left (1, \frac {\left (-2 \sqrt {x}\, a b +\left (b^{2}+x \right ) a -y\right ) 2^{{1}/{3}}}{2 \left (b^{2}\right )^{{1}/{3}} a}\right ) b} = 0
\]
Maple trace
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
differential order: 1; looking for linear symmetries
trying exact
trying Abel
<- Abel successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-a \left (1-\frac {b}{\sqrt {x}}\right ) y \left (x \right )=\frac {a^{2} b}{\sqrt {x}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {a \left (1-\frac {b}{\sqrt {x}}\right ) y \left (x \right )+\frac {a^{2} b}{\sqrt {x}}}{y \left (x \right )} \end {array} \]
2.24.11.2 ✓ Mathematica. Time used: 1.029 (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]
\[
\text {Solve}\left [\frac {\sqrt [3]{-1} 2^{2/3} \sqrt [3]{\left (b-\sqrt {x}\right )^3} \operatorname {AiryAi}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} \left (\left (b-\sqrt {x}\right )^3\right )^{2/3} \left (a \left (b-\sqrt {x}\right )^2-y(x)\right )}{a b^{2/3} \left (b-\sqrt {x}\right )^2}\right )-2 \sqrt [3]{b} \operatorname {AiryAiPrime}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} \left (\left (b-\sqrt {x}\right )^3\right )^{2/3} \left (a \left (b-\sqrt {x}\right )^2-y(x)\right )}{a b^{2/3} \left (b-\sqrt {x}\right )^2}\right )}{\sqrt [3]{-1} 2^{2/3} \sqrt [3]{\left (b-\sqrt {x}\right )^3} \operatorname {AiryBi}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} \left (\left (b-\sqrt {x}\right )^3\right )^{2/3} \left (a \left (b-\sqrt {x}\right )^2-y(x)\right )}{a b^{2/3} \left (b-\sqrt {x}\right )^2}\right )-2 \sqrt [3]{b} \operatorname {AiryBiPrime}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} \left (\left (b-\sqrt {x}\right )^3\right )^{2/3} \left (a \left (b-\sqrt {x}\right )^2-y(x)\right )}{a b^{2/3} \left (b-\sqrt {x}\right )^2}\right )}+c_1=0,y(x)\right ]
\]
2.24.11.3 ✗ 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