Internal problem ID [10797]
Internal file name [OUTPUT/9745_Monday_June_06_2022_09_31_26_PM_14092097/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. subsection 1.3.3-2. Equations
of the form \(y y'=f_1(x) y+f_0(x)\)
Problem number: 61.
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 A`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-a \left (\left (-3+2 k \right ) x +1\right ) x^{-k} y=a^{2} \left (k -2\right ) \left (\left (k -1\right ) x +1\right ) x^{-2 k +2}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-a \left (\left (-3+2 k \right ) x +1\right ) x^{-k} y=a^{2} \left (k -2\right ) \left (\left (k -1\right ) x +1\right ) x^{-2 k +2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a \left (\left (-3+2 k \right ) x +1\right ) x^{-k} y+a^{2} \left (k -2\right ) \left (\left (k -1\right ) x +1\right ) x^{-2 k +2}}{y} \end {array} \]
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)*(2*k^2*x-5*k*x+k+3*x)/(x*(2*k*x-3*x+1)), 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)-(-1+k)*(2*k*x-3*x+2)*y(x)/(x*(k*x-x+1)), y(x)` *** Sublevel 2 *** 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+k)*(2*k*x-3*x+2)*y(x)/(x*(k*x-x+1)), 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)-a*((2*k-3)*x+1)*x^(-k)*y(x)=a^2*(k-2)*((k-1)*x+1)*x^(2*(1-k)),y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-a*((2*k-3)*x+1)*x^(-k)*y[x]==a^2*(k-2)*((k-1)*x+1)*x^(2*(1-k)),y[x],x,IncludeSingularSolutions -> True]
Not solved