2.22.48 Problem 56

Internal problem ID [13543]
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 : 56
Date solved : Wednesday, December 31, 2025 at 09:56:16 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

Entering first order ode abel second kind solver

\begin{align*} y^{\prime } y-y&=\frac {15 x}{4}+\frac {A}{x^{7}} \\ \end{align*}

2.22.48.1 Solved using first_order_ode_abel_second_kind_table_5_lookup

0.143 (sec)

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

\[ y = \frac {a \left (-3 I_{1}^{2}+I_{2} I_{3} \right )}{2 I_{1}^{{3}/{2}} I_{3}^{{3}/{8}}} \]

Where

\begin{align*} x&=\frac {a \sqrt {I_{1}}}{I_{3}^{{3}/{8}}}\\ I_{1}&=t \left (2 \int \frac {t}{4 t^{3}-1}d t -\frac {\sqrt {4 t^{3}-1}}{t}+c_1 \right )\\ I_{2}&=\frac {\sqrt {4 t^{3}-1}\, t \left (2 \int \frac {t}{4 t^{3}-1}d t -\frac {\sqrt {4 t^{3}-1}}{t}+c_1 \right )-1}{t}\\ I_{3}&=4 t^{3} \left (2 \int \frac {t}{4 t^{3}-1}d t -\frac {\sqrt {4 t^{3}-1}}{t}+c_1 \right )^{2}-\frac {\left (\sqrt {4 t^{3}-1}\, t \left (2 \int \frac {t}{4 t^{3}-1}d t -\frac {\sqrt {4 t^{3}-1}}{t}+c_1 \right )-1\right )^{2}}{t^{2}}\\ a&=\frac {4^{{1}/{8}} 3^{{7}/{8}} A^{{1}/{8}}}{3} \end{align*}

Summary of solutions found

\begin{align*} \left (y&=\frac {a \left (-3 I_{1}^{2}+I_{2} I_{3} \right )}{2 I_{1}^{{3}/{2}} I_{3}^{{3}/{8}}}\right )\boldsymbol {\operatorname {where}}\left [x =\frac {a \sqrt {I_{1}}}{I_{3}^{{3}/{8}}}, I_{1} =t \left (2 \int \frac {t}{4 t^{3}-1}d t -\frac {\sqrt {4 t^{3}-1}}{t}+c_1 \right ), I_{2} =\frac {\sqrt {4 t^{3}-1}\, t \left (2 \int \frac {t}{4 t^{3}-1}d t -\frac {\sqrt {4 t^{3}-1}}{t}+c_1 \right )-1}{t}, I_{3} =4 t^{3} \left (2 \int \frac {t}{4 t^{3}-1}d t -\frac {\sqrt {4 t^{3}-1}}{t}+c_1 \right )^{2}-\frac {\left (\sqrt {4 t^{3}-1}\, t \left (2 \int \frac {t}{4 t^{3}-1}d t -\frac {\sqrt {4 t^{3}-1}}{t}+c_1 \right )-1\right )^{2}}{t^{2}}, a =\frac {4^{{1}/{8}} 3^{{7}/{8}} A^{{1}/{8}}}{3}\right ] \\ \end{align*}

2.22.48.2 ✗ Maple

ode:=y(x)*diff(y(x),x)-y(x) = 15/4*x+A/x^7; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

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 
Looking for potential symmetries 
trying inverse_Riccati 
trying an equivalence to an Abel ODE 
differential order: 1; trying a linearization to 2nd order 
--- trying a change of variables {x -> y(x), y(x) -> x} 
differential order: 1; trying a linearization to 2nd order 
trying 1st order ODE linearizable_by_differentiation 
--- Trying Lie symmetry methods, 1st order --- 
   -> Computing symmetries using: way = 3 
   -> Computing symmetries using: way = 4 
   -> Computing symmetries using: way = 2 
trying symmetry patterns for 1st order ODEs 
-> trying a symmetry pattern of the form [F(x)*G(y), 0] 
-> trying a symmetry pattern of the form [0, F(x)*G(y)] 
-> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] 
-> trying a symmetry pattern of the form [F(x),G(x)] 
-> trying a symmetry pattern of the form [F(y),G(y)] 
-> trying a symmetry pattern of the form [F(x)+G(y), 0] 
-> trying a symmetry pattern of the form [0, F(x)+G(y)] 
-> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] 
-> trying a symmetry pattern of conformal type
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=\frac {15 x}{4}+\frac {A}{x^{7}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )+\frac {15 x}{4}+\frac {A}{x^{7}}}{y \left (x \right )} \end {array} \]

2.22.48.3 ✗ Mathematica

ode=y[x]*D[y[x],x]-y[x]==15/4*x+A*x^(-7); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.22.48.4 ✗ Sympy

from sympy import * 
x = symbols("x") 
A = symbols("A") 
y = Function("y") 
ode = Eq(-A/x**7 - 15*x/4 + y(x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -A/(x**7*y(x)) - 15*x/(4*y(x)) + Derivative(y(x), x) - 1 cannot be solved by the factorable group method