2.22.28 Problem 33

Internal problem ID [13523]
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 : 33
Date solved : Wednesday, December 31, 2025 at 09:54:10 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

Entering first order ode abel second kind solver

\begin{align*} y^{\prime } y-y&=\frac {A}{x^{2}} \\ \end{align*}

2.22.28.1 Solved using first_order_ode_abel_second_kind_table_5_lookup

0.181 (sec)

This solution is given by a lookup into table 5 in the book Handbook of exact solutions in parametric form as follows

\[ y = \frac {3 a U_{3}}{t^{{2}/{3}} Z \left (t \right ) U_{2}} \]

Where

\begin{align*} Z \left (t \right )&=c_1 \operatorname {BesselJ}\left (\frac {1}{3}, t\right )+c_2 \operatorname {BesselY}\left (\frac {1}{3}, t\right )\\ x&=\frac {2 a \,t^{{4}/{3}} Z \left (t \right )^{2}}{U_{2}}\\ a&=\left (-\frac {A}{36}\right )^{{1}/{3}}\\ U_{1}&=t \left (\frac {d}{d t}Z \left (t \right )\right )+\frac {Z \left (t \right )}{3}\\ U_{2}&=U_{1}^{2}+t^{2} Z \left (t \right )^{2}\\ U_{3}&=\frac {2 t^{2} Z \left (t \right )^{3}}{3}-2 U_{2} U_{1} \end{align*}

Summary of solutions found

\begin{align*} \left (y&=\frac {3 a U_{3}}{t^{{2}/{3}} Z \left (t \right ) U_{2}}\right )\boldsymbol {\operatorname {where}}\left \{U_{1} =t \left (\frac {d}{d t}Z \left (t \right )\right )+\frac {Z \left (t \right )}{3}, U_{2} =U_{1}^{2}+t^{2} Z \left (t \right )^{2}, U_{3} =\frac {2 t^{2} Z \left (t \right )^{3}}{3}-2 U_{2} U_{1} , a =\left (-\frac {A}{36}\right )^{{1}/{3}}, x =\frac {2 a \,t^{{4}/{3}} Z \left (t \right )^{2}}{U_{2}}, Z \left (t \right )=c_1 \operatorname {BesselJ}\left (\frac {1}{3}, t\right )+c_2 \operatorname {BesselY}\left (\frac {1}{3}, t\right )\right \} \\ \end{align*}

2.22.28.2 ✓ Maple. Time used: 0.003 (sec). Leaf size: 277

ode:=y(x)*diff(y(x),x)-y(x) = A/x^2; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {A \left (\operatorname {AiryBi}\left (-\frac {\left (x^{3}-2 y x^{2}+x y^{2}+2 A \right ) 2^{{2}/{3}}}{4 \left (-A^{2}\right )^{{1}/{3}} x}\right ) c_1 -\operatorname {AiryAi}\left (-\frac {\left (x^{3}-2 y x^{2}+x y^{2}+2 A \right ) 2^{{2}/{3}}}{4 \left (-A^{2}\right )^{{1}/{3}} x}\right )\right ) \left (x -y\right ) 2^{{1}/{3}}-2 \left (\operatorname {AiryBi}\left (1, -\frac {\left (x^{3}-2 y x^{2}+x y^{2}+2 A \right ) 2^{{2}/{3}}}{4 \left (-A^{2}\right )^{{1}/{3}} x}\right ) c_1 -\operatorname {AiryAi}\left (1, -\frac {\left (x^{3}-2 y x^{2}+x y^{2}+2 A \right ) 2^{{2}/{3}}}{4 \left (-A^{2}\right )^{{1}/{3}} x}\right )\right ) \left (-A^{2}\right )^{{2}/{3}}}{A 2^{{1}/{3}} \left (x -y\right ) \operatorname {AiryBi}\left (-\frac {\left (x^{3}-2 y x^{2}+x y^{2}+2 A \right ) 2^{{2}/{3}}}{4 \left (-A^{2}\right )^{{1}/{3}} x}\right )-2 \operatorname {AiryBi}\left (1, -\frac {\left (x^{3}-2 y x^{2}+x y^{2}+2 A \right ) 2^{{2}/{3}}}{4 \left (-A^{2}\right )^{{1}/{3}} x}\right ) \left (-A^{2}\right )^{{2}/{3}}} = 0 \]

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

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=\frac {A}{x^{2}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )+\frac {A}{x^{2}}}{y \left (x \right )} \end {array} \]

2.22.28.3 ✓ Mathematica. Time used: 2.595 (sec). Leaf size: 213

ode=y[x]*D[y[x],x]-y[x]==A*x^(-2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {\operatorname {AiryAiPrime}\left (-\frac {\sqrt [3]{-\frac {1}{2}} \left (x^3-2 y(x) x^2+y(x)^2 x+2 A\right )}{2 A^{2/3} x}\right )-\frac {\left (-\frac {1}{2}\right )^{2/3} (x-y(x)) \operatorname {AiryAi}\left (-\frac {\sqrt [3]{-\frac {1}{2}} \left (x^3-2 y(x) x^2+y(x)^2 x+2 A\right )}{2 A^{2/3} x}\right )}{\sqrt [3]{A}}}{\operatorname {AiryBiPrime}\left (-\frac {\sqrt [3]{-\frac {1}{2}} \left (x^3-2 y(x) x^2+y(x)^2 x+2 A\right )}{2 A^{2/3} x}\right )-\frac {\left (-\frac {1}{2}\right )^{2/3} (x-y(x)) \operatorname {AiryBi}\left (-\frac {\sqrt [3]{-\frac {1}{2}} \left (x^3-2 y(x) x^2+y(x)^2 x+2 A\right )}{2 A^{2/3} x}\right )}{\sqrt [3]{A}}}+c_1=0,y(x)\right ] \]

2.22.28.4 ✗ Sympy

from sympy import * 
x = symbols("x") 
A = symbols("A") 
y = Function("y") 
ode = Eq(-A/x**2 + y(x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -A/(x**2*y(x)) + Derivative(y(x), x) - 1 cannot be solved by the factorable group method