Internal problem ID [10720]
Internal file name [OUTPUT/9668_Monday_June_06_2022_03_21_16_PM_27444372/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: 72.
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.
\[ \boxed {y y^{\prime }-y=-\frac {\left (1+m \right ) x}{\left (2+m \right )^{2}}+A \,x^{1+2 m}+B \,x^{3 m +1}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & A \,x^{1+2 m} m^{2}+B \,x^{3 m +1} m^{2}-y y^{\prime } m^{2}+4 A \,x^{1+2 m} m +4 B \,x^{3 m +1} m -4 y y^{\prime } m +y m^{2}+4 A \,x^{1+2 m}+4 B \,x^{3 m +1}-4 y y^{\prime }+4 y m -m x +4 y-x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-A \,x^{1+2 m} m^{2}-B \,x^{3 m +1} m^{2}-4 A \,x^{1+2 m} m -4 B \,x^{3 m +1} m -y m^{2}-4 A \,x^{1+2 m}-4 B \,x^{3 m +1}-4 y m +m x -4 y+x}{-y m^{2}-4 y m -4 y} \end {array} \]
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*A*x^(1+2*m)*m^3+3*B*x^(1+3*m)*m^3+9*A*x^(1+2*m)*m^2+13*B*x^(1+3*m)*m^2+12 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)+K[1]*(1+m)/(x*(m+2)^2), y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature 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)=-(m+1)/(m+2)^2*x+A*x^(2*m+1)+B*x^(3*m+1),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]==-(m+1)/(m+2)^2*x+A*x^(2*m+1)+B*x^(3*m+1),y[x],x,IncludeSingularSolutions -> True]
Not solved