Internal problem ID [10660]
Internal file name [OUTPUT/9608_Monday_June_06_2022_03_13_31_PM_8547601/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: 12.
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.
\[ \boxed {y y^{\prime }-y=\frac {2 m -2}{\left (m -3\right )^{2}}+\frac {2 A \left (m \left (m +3\right ) \sqrt {x}+\left (4 m^{2}+3 m +9\right ) A +\frac {3 m \left (m +3\right ) A^{2}}{\sqrt {x}}\right )}{\left (m -3\right )^{2}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 6 A^{3} m^{2}+8 A^{2} \sqrt {x}\, m^{2}+2 A x \,m^{2}-y y^{\prime } \sqrt {x}\, m^{2}+18 A^{3} m +6 A^{2} \sqrt {x}\, m +6 A x m +6 y y^{\prime } \sqrt {x}\, m +y \sqrt {x}\, m^{2}+18 A^{2} \sqrt {x}-9 y y^{\prime } \sqrt {x}-6 y \sqrt {x}\, m +9 y \sqrt {x}+2 \sqrt {x}\, m -2 \sqrt {x}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-6 A^{3} m^{2}-8 A^{2} \sqrt {x}\, m^{2}-2 A x \,m^{2}-18 A^{3} m -6 A^{2} \sqrt {x}\, m -6 A x m -y \sqrt {x}\, m^{2}-18 A^{2} \sqrt {x}+6 y \sqrt {x}\, m -9 y \sqrt {x}-2 \sqrt {x}\, m +2 \sqrt {x}}{-y \sqrt {x}\, m^{2}+6 y \sqrt {x}\, m -9 y \sqrt {x}} \end {array} \]
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)-(1/2)*A*y(x)*m*(3*A^2-x)*(m+3)/(4*A^2*x^(3/2)*m^2+3*A^3*m^2*x+3*A^2*x^(3/2)*m+9*A 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)+(1/12)*(12*A^3*y(x)*m^2+36*A^3*y(x)*m-8*A^2*m^2*x-6*A^2*m*x-x^2*m^2-18*A^2*x+6*m* 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*(4*A^2*y(x)*m^2+3*A^2*y(x)*m-A*m^2*x+9*A^2*y(x)-3*A*m*x+y(x)*m-y(x))/(x*(8*A^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)=2*(m-1)/(m-3)^2+2*A/(m-3)^2*(m*(m+3)*x^(1/2)+(4*m^2+3*m+9)*A+3*m*(m+3)*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]==2*(m-1)/(m-3)^2+2*A/(m-3)^2*(m*(m+3)*x^(1/2)+(4*m^2+3*m+9)*A+3*m*(m+3)*A^2*x^(-1/2)),y[x],x,IncludeSingularSolutions -> True]
Not solved