2.22.46 Problem 54

Internal problem ID [13541]
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 : 54
Date solved : Wednesday, December 31, 2025 at 09:55:43 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

Entering first order ode abel second kind solver

\begin{align*} y^{\prime } y-y&=6 x +\frac {A}{x^{4}} \\ \end{align*}

2.22.46.1 Solved using first_order_ode_abel_second_kind_table_5_lookup

0.089 (sec)

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

\[ y = \frac {a \left (5 \sqrt {4 t^{3}-1}\, I_{1} -2\right )}{t^{{3}/{5}} I_{1}^{{2}/{5}}} \]

Where

\begin{align*} x&=\frac {a}{t^{{3}/{5}} I_{1}^{{2}/{5}}}\\ I_{1}&=t \left (2 \int \frac {t}{4 t^{3}-1}d t -\frac {\sqrt {4 t^{3}-1}}{t}+c_1 \right )\\ a&=\frac {150^{{4}/{5}} A^{{1}/{5}}}{150} \end{align*}

Summary of solutions found

\begin{align*} \left (y&=\frac {a \left (5 \sqrt {4 t^{3}-1}\, I_{1} -2\right )}{t^{{3}/{5}} I_{1}^{{2}/{5}}}\right )\boldsymbol {\operatorname {where}}\left [x =\frac {a}{t^{{3}/{5}} I_{1}^{{2}/{5}}}, I_{1} =t \left (2 \int \frac {t}{4 t^{3}-1}d t -\frac {\sqrt {4 t^{3}-1}}{t}+c_1 \right ), a =\frac {150^{{4}/{5}} A^{{1}/{5}}}{150}\right ] \\ \end{align*}

2.22.46.2 ✓ Maple. Time used: 0.002 (sec). Leaf size: 217

ode:=y(x)*diff(y(x),x)-y(x) = 6*x+A/x^4; 
dsolve(ode,y(x), singsol=all);
 

\[ c_1 +\frac {5 \,5^{{2}/{3}} \left (\left (\frac {12 x^{5}+12 y x^{4}+3 x^{3} y^{2}+2 A}{x^{9} \left (2 x +y\right )^{6}}\right )^{{1}/{6}} 3^{{5}/{6}} A x \operatorname {hypergeom}\left (\left [\frac {1}{6}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -\frac {2 A}{3 x^{3} \left (2 x +y\right )^{2}}\right )+\frac {24 \left (6 x^{5}+6 y x^{4}+\frac {3 x^{3} y^{2}}{2}+A \right ) \left (\frac {\left (-\frac {1}{x^{{3}/{2}} \left (2 x +y\right )}\right )^{{2}/{3}} y}{6}+\left (-\frac {1}{x^{{3}/{2}} \left (2 x +y\right )}\right )^{{5}/{3}} x^{{5}/{2}} \left (x +\frac {y}{2}\right )\right )}{5}\right ) \left (x +\frac {y}{2}\right )}{2 \left (-\frac {1}{x^{{3}/{2}} \left (2 x +y\right )}\right )^{{7}/{3}} \left (\frac {12 x^{5}+12 y x^{4}+3 x^{3} y^{2}+2 A}{x^{3} \left (2 x +y\right )^{2}}\right )^{{1}/{6}} x^{{11}/{2}} \left (2 x +y\right )^{4}} = 0 \]

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 
found: 2 potential symmetries. Proceeding with integration step 
<- Abel successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=6 x +\frac {A}{x^{4}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )+6 x +\frac {A}{x^{4}}}{y \left (x \right )} \end {array} \]

2.22.46.3 ✓ Mathematica. Time used: 0.878 (sec). Leaf size: 213

ode=y[x]*D[y[x],x]-y[x]==6*x+A*x^(-4); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [c_1=\frac {i \left (-\frac {2 A+12 x^5+12 x^4 y(x)+3 x^3 y(x)^2}{A}\right )^{5/6} \left (-10\ 2^{5/6} x^5 \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {3}{2},-\frac {3 x^3 (2 x+y(x))^2}{2 A}\right )-5\ 2^{5/6} x^4 y(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {3}{2},-\frac {3 x^3 (2 x+y(x))^2}{2 A}\right )+A \left (\frac {2 A+12 x^5+12 x^4 y(x)+3 x^3 y(x)^2}{A}\right )^{5/6}\right )}{2 \sqrt [3]{2} \sqrt {3} \sqrt {A} x^{5/2} \left (\frac {2 A+12 x^5+12 x^4 y(x)+3 x^3 y(x)^2}{A}\right )^{5/6}},y(x)\right ] \]

2.22.46.4 ✗ Sympy

from sympy import * 
x = symbols("x") 
A = symbols("A") 
y = Function("y") 
ode = Eq(-A/x**4 - 6*x + y(x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -A/(x**4*y(x)) - 6*x/y(x) + Derivative(y(x), x) - 1 cannot be solved by the factorable group method