2.22.40 Problem 47

Internal problem ID [13535]
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 : 47
Date solved : Wednesday, December 31, 2025 at 09:54:51 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

Entering first order ode abel second kind solver

\begin{align*} y^{\prime } y-y&=12 x +\frac {A}{x^{{5}/{2}}} \\ \end{align*}

2.22.40.1 Solved using first_order_ode_abel_second_kind_table_5_lookup

0.223 (sec)

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

\[ y = \frac {a \left (14 p^{2} E_{4} -3\right )}{p^{{6}/{7}} E_{3}^{{4}/{7}}} \]

Where

\begin{align*} x&=\frac {a}{p^{{6}/{7}} E_{3}^{{4}/{7}}}\\ t&=\int \frac {1}{\sqrt {4 p^{3}-1}}d p -c_1\\ p&=\operatorname {RootOf}\left (-\int \frac {1}{\sqrt {4 \textit {\_Z}^{3}-1}}d \textit {\_Z} +c_1 +t \right )\\ E_{3}&=t^{3} \sqrt {4 p^{3}-1}+2 t \,p^{2}\\ E_{4}&=t \sqrt {4 p^{3}-1}+2 p \end{align*}

Summary of solutions found

\begin{align*} \left (y&=\frac {a \left (14 p^{2} E_{4} -3\right )}{p^{{6}/{7}} E_{3}^{{4}/{7}}}\right )\boldsymbol {\operatorname {where}}\left [x =\frac {a}{p^{{6}/{7}} E_{3}^{{4}/{7}}}, E_{3} =t^{3} \sqrt {4 p^{3}-1}+2 t \,p^{2}, E_{4} =t \sqrt {4 p^{3}-1}+2 p , t =\int \frac {1}{\sqrt {4 p^{3}-1}}d p -c_1 , a =\frac {147^{{5}/{7}} A^{{2}/{7}}}{147}, p =\operatorname {RootOf}\left (-\int \frac {1}{\sqrt {4 \textit {\_Z}^{3}-1}}d \textit {\_Z} +c_1 +t \right )\right ] \\ \end{align*}

2.22.40.2 ✓ Maple. Time used: 0.002 (sec). Leaf size: 110

ode:=y(x)*diff(y(x),x)-y(x) = 12*x+A/x^(5/2); 
dsolve(ode,y(x), singsol=all);
 

\[ c_1 +\frac {12 \left (2^{{2}/{3}} \left (\frac {3 x^{{3}/{2}} y^{2}}{4}-6 y x^{{5}/{2}}+A +12 x^{{7}/{2}}\right ) \left (\frac {48 x^{{7}/{2}}-24 y x^{{5}/{2}}+3 x^{{3}/{2}} y^{2}+4 A}{A}\right )^{{1}/{6}}-56 \operatorname {hypergeom}\left (\left [-\frac {1}{6}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -\frac {3 x^{{3}/{2}} \left (-4 x +y\right )^{2}}{4 A}\right ) x^{{5}/{2}} \left (x -\frac {y}{4}\right )\right ) \sqrt {3}}{\sqrt {-A \,x^{{7}/{2}}}} = 0 \]

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 
   Looking for potential symmetries 
   Looking for potential symmetries 
<- Abel successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=12 x +\frac {A}{x^{{5}/{2}}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )+12 x +\frac {A}{x^{{5}/{2}}}}{y \left (x \right )} \end {array} \]

2.22.40.3 ✗ Mathematica

ode=y[x]*D[y[x],x]-y[x]==12*x+A*x^(-5/2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.22.40.4 ✗ Sympy

from sympy import * 
x = symbols("x") 
A = symbols("A") 
y = Function("y") 
ode = Eq(-A/x**(5/2) - 12*x + y(x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out