Internal problem ID [10688]
Internal file name [OUTPUT/9636_Monday_June_06_2022_03_17_06_PM_3102030/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: 40.
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.
\[ \boxed {y y^{\prime }-y=-\frac {3 x}{16}+\frac {3 A}{x^{\frac {1}{3}}}-\frac {12 A^{2}}{x^{\frac {5}{3}}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 16 y y^{\prime } x^{\frac {5}{3}}-16 y x^{\frac {5}{3}}+3 x^{\frac {8}{3}}-48 A \,x^{\frac {4}{3}}+192 A^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-3 x^{\frac {8}{3}}+16 y x^{\frac {5}{3}}+48 A \,x^{\frac {4}{3}}-192 A^{2}}{16 y x^{\frac {5}{3}}} \end {array} \]
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 Looking for potential symmetries Looking for potential symmetries trying Riccati to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = -(4*x-3)*(diff(y(x), x))/(x*(x-1))-2*y(x)/((x-1)*x), y(x)` *** Sublevel Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] <- linear_1 successful <- Riccati to 2nd Order successful <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 1488
dsolve(y(x)*diff(y(x),x)-y(x)=-3/16*x+3*A*x^(-1/3)-12*A^2*x^(-5/3),y(x), singsol=all)
\[ \text {Expression too large to display} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==-3/16*x+3*A*x^(-1/3)-12*A^2*x^(-5/3),y[x],x,IncludeSingularSolutions -> True]
Not solved