Internal problem ID [10668]
Internal file name [OUTPUT/9616_Monday_June_06_2022_03_15_55_PM_70549492/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: 20.
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.
\[ \boxed {y y^{\prime }-y=-\frac {6 X}{25}+\frac {2 A \left (2 \sqrt {x}+19 A +\frac {6 A^{2}}{\sqrt {x}}\right )}{25}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 12 A^{3}+38 A^{2} \sqrt {x}+4 A x -25 y y^{\prime } \sqrt {x}+25 y \sqrt {x}-6 X \sqrt {x}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-12 A^{3}-38 A^{2} \sqrt {x}-4 A x -25 y \sqrt {x}+6 X \sqrt {x}}{25 y \sqrt {x}} \end {array} \]
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 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)] `, `-> Computing symmetries using: way = HINT -> Calling odsolve with the ODE`, diff(y(x), x), y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)-A*y(x)*(3*A^2-x)/(19*A^2*x^(3/2)+6*A^3*x-3*x^(3/2)*X+2*x^2*A), y(x)` *** Sub Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful `, `-> Computing symmetries using: way = HINT -> Calling odsolve with the ODE`, diff(y(x), x)+(1/24)*(24*A^3*y(x)-38*A^2*x+6*X*x-25*x^2)/(A^3*x), y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)+2*(19*A^2*y(x)-2*A*x-3*y(x)*X)/(x*(38*A^2-6*X+25*x)), y(x)` *** Sublevel 2 * Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> 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`
✗ Solution by Maple
dsolve(y(x)*diff(y(x),x)-y(x)=-6/25*X+2/25*A*(2*x^(1/2)+19*A+6*A^2*x^(-1/2)),y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==-6/25*X+2/25*A*(2*x^(1/2)+19*A+6*A^2*x^(-1/2)),y[x],x,IncludeSingularSolutions -> True]
Not solved