Internal problem ID [10821]
Internal file name [OUTPUT/9803_Sunday_June_19_2022_09_25_21_PM_56552507/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 1.3.4-2. Equations
of the form \((g_1(x)+g_0(x))y'=f_2(x) y^2+f_1(x) y+f_0(x)\)
Problem number: 5.
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.
\[ \boxed {\left (y+a \,x^{n +1}+b \,x^{n}\right ) y^{\prime }-\left (a n \,x^{n}+c \,x^{n -1}\right ) y=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (y+a \,x^{n +1}+b \,x^{n}\right ) y^{\prime }-\left (a n \,x^{n}+c \,x^{n -1}\right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\left (a n \,x^{n}+c \,x^{n -1}\right ) y}{y+a \,x^{n +1}+b \,x^{n}} \end {array} \]
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)+(x^n*y(x)*a*n^2+x^(n-1)*y(x)*c*n-x^n*a*n*x-x^(n-1)*y(x)*c-x^(n-1)*c*x)/(x*(a*n*x^ 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)-y(x)*(x^(n+1)*x^n*a^2*n+2*x^(n+1)*x^(n-1)*a*c+x^n*x^(n-1)*b*c)/((x^(n+1)*a+b*x^n) 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), 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)-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 -> 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)+a*x^(n+1)+b*x^n)*diff(y(x),x)=(a*n*x^n+c*x^(n-1))*y(x),y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[(y[x]+a*x^(n+1)+b*x^n)*y'[x]==(a*n*x^n+c*x^(n-1))*y[x],y[x],x,IncludeSingularSolutions -> True]
Not solved