2.22.16 Problem 19

Internal problem ID [13511]
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 : 19
Date solved : Wednesday, December 31, 2025 at 09:53:24 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

Entering first order ode abel second kind solver

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

2.22.16.1 Solved using first_order_ode_abel_second_kind_table_5_lookup

0.238 (sec)

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

\[ y = \frac {a \left (\frac {2 t}{3}-R E \right )}{E^{{2}/{3}}} \]

Where

\begin{align*} x&=\frac {a t}{3 E^{{2}/{3}}}\\ a&=\left (-\frac {2 A}{243}\right )^{{1}/{3}}\\ E&=\sqrt {t \left (t +1\right )}-\ln \left ({| c_1 \left (\sqrt {t}+\sqrt {t +1}\right )|}\right )\\ R&=\sqrt {\frac {t +1}{t}} \end{align*}

Summary of solutions found

\begin{align*} \left (y&=\frac {a \left (\frac {2 t}{3}-R E \right )}{E^{{2}/{3}}}\right )\boldsymbol {\operatorname {where}}\left \{E =\sqrt {t \left (t +1\right )}-\ln \left ({| c_1 \left (\sqrt {t}+\sqrt {t +1}\right )|}\right ), R =\sqrt {\frac {t +1}{t}}, a =\left (-\frac {2 A}{243}\right )^{{1}/{3}}, x =\frac {a t}{3 E^{{2}/{3}}}\right \} \\ \end{align*}

2.22.16.2 ✓ Maple. Time used: 0.001 (sec). Leaf size: 170

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

\[ \frac {6 x \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {\left (A^{2}\right )^{{1}/{3}} x}{A}}\, \left (-2 x +y\right )}{\sqrt {\frac {\left (A^{2}\right )^{{1}/{3}} \left (4 x^{3}-4 y x^{2}+x y^{2}+2 A \right )}{A y^{2}}}\, y}\right ) A +\frac {c_1}{6}\right ) \sqrt {\frac {\left (A^{2}\right )^{{1}/{3}} x}{A}}+2 y \sqrt {3}\, \left (-x^{3}-\frac {y x^{2}}{2}+\frac {x y^{2}}{2}+A \right ) \sqrt {\frac {\left (A^{2}\right )^{{1}/{3}} \left (4 x^{3}-4 y x^{2}+x y^{2}+2 A \right )}{A y^{2}}}}{\sqrt {\frac {\left (A^{2}\right )^{{1}/{3}} x}{A}}\, x} = 0 \]

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

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

2.22.16.3 ✓ Mathematica. Time used: 0.856 (sec). Leaf size: 233

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

\[ \text {Solve}\left [c_1=-\frac {i \sqrt {-\frac {2 A+4 x^3-4 x^2 y(x)+x y(x)^2}{A}} \left (-6 \sqrt {A} x^{3/2} \text {arcsinh}\left (\frac {\sqrt {x} (2 x-y(x))}{\sqrt {2} \sqrt {A}}\right )+x^2 (-y(x)) \sqrt {\frac {2 A+4 x^3-4 x^2 y(x)+x y(x)^2}{A}}+x y(x)^2 \sqrt {\frac {2 A+4 x^3-4 x^2 y(x)+x y(x)^2}{A}}+2 \left (A-x^3\right ) \sqrt {\frac {2 A+4 x^3-4 x^2 y(x)+x y(x)^2}{A}}\right )}{4 \sqrt {A} x^{3/2} \sqrt {\frac {2 A+4 x^3-4 x^2 y(x)+x y(x)^2}{A}}},y(x)\right ] \]

2.22.16.4 ✗ Sympy

from sympy import * 
x = symbols("x") 
A = symbols("A") 
y = Function("y") 
ode = Eq(-A/x**2 - 2*x + 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)) - 2*x/y(x) + Derivative(y(x), x) - 1 cannot be solved by the factorable group method