22.23 problem 23

Internal problem ID [10671]
Internal file name [OUTPUT/9619_Monday_June_06_2022_03_16_07_PM_474942/index.tex]

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. Form \(y y'-y=f(x)\). subsection 1.3.1-2. Solvable equations and their solutions
Problem number: 23.
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program : "unknown"

Maple gives the following as the ode type

[_rational, [_Abel, `2nd type`, `class B`]]

Unable to solve or complete the solution.

\[ \boxed {y y^{\prime }-y=-\frac {9 x}{100}+\frac {A}{x^{\frac {5}{3}}}} \] Unable to determine ODE type.

22.23.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -100 y y^{\prime } x^{\frac {5}{3}}+100 y x^{\frac {5}{3}}-9 x^{\frac {8}{3}}+100 A =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-100 y x^{\frac {5}{3}}+9 x^{\frac {8}{3}}-100 A}{100 y x^{\frac {5}{3}}} \end {array} \]

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`
 

✓ Solution by Maple

Time used: 0.0 (sec). Leaf size: 581

dsolve(y(x)*diff(y(x),x)-y(x)=-9/100*x+A*x^(-5/3),y(x), singsol=all)
 

\[ \text {Expression too large to display} \]

✓ Solution by Mathematica

Time used: 60.566 (sec). Leaf size: 7909

DSolve[y[x]*y'[x]-y[x]==-9/100*x+A*x^(-5/3),y[x],x,IncludeSingularSolutions -> True]
 

Too large to display