Internal problem ID [10800]
Internal file name [OUTPUT/9748_Wednesday_June_08_2022_05_53_20_PM_36010500/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: 64.
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.
\[ \boxed {y y^{\prime }-\frac {a \left (\frac {\left (n +4\right ) x}{n +2}-2\right ) x^{-\frac {1+2 n}{n}} y}{n}=\frac {a^{2} \left (2 x^{2}+\left (n^{2}+n -4\right ) x -\left (n -1\right ) \left (n +2\right )\right ) x^{-\frac {2+3 n}{n}}}{n \left (n +2\right )}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{-\frac {2+3 n}{n}} a^{2} n^{2} x -a^{2} x^{-\frac {2+3 n}{n}} n^{2}+x^{-\frac {2+3 n}{n}} a^{2} n x +2 x^{-\frac {2+3 n}{n}} a^{2} x^{2}+y x^{-\frac {1+2 n}{n}} a n x -x^{-\frac {2+3 n}{n}} a^{2} n -4 x^{-\frac {2+3 n}{n}} a^{2} x -2 y x^{-\frac {1+2 n}{n}} a n +4 y x^{-\frac {1+2 n}{n}} a x -y y^{\prime } n^{2}+2 a^{2} x^{-\frac {2+3 n}{n}}-4 a \,x^{-\frac {1+2 n}{n}} y-2 y y^{\prime } n =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-x^{-\frac {2+3 n}{n}} a^{2} n^{2} x -y x^{-\frac {1+2 n}{n}} a n x +a^{2} x^{-\frac {2+3 n}{n}} n^{2}-x^{-\frac {2+3 n}{n}} a^{2} n x -2 x^{-\frac {2+3 n}{n}} a^{2} x^{2}+2 y x^{-\frac {1+2 n}{n}} a n -4 y x^{-\frac {1+2 n}{n}} a x +x^{-\frac {2+3 n}{n}} a^{2} n +4 x^{-\frac {2+3 n}{n}} a^{2} x +4 a \,x^{-\frac {1+2 n}{n}} y-2 a^{2} x^{-\frac {2+3 n}{n}}}{-y n^{2}-2 n y} \end {array} \]
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)*(n^2*x-4*n^2+5*n*x-10*n+4*x-4)/(n*x*(n*x-2*n+4*x-4)), y(x)` *** Subleve 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)*(2*n^3*x-3*n^3+4*n^2*x+2*n*x^2-5*n^2-6*n*x+4*x^2+4*n-8*x+4)/(x*n*(x-1)*(n^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 -> 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/n*((n+4)/(n+2)*x-2)*x^(-(2*n+1)/n)*y(x)=a^2/(n*(n+2))*(2*x^2+(n^2+n-4)*x-(n-1)*(n+2))*x^(-(3*n+2)/n),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/n*((n+4)/(n+2)*x-2)*x^(-(2*n+1)/n)*y[x]==a^2/(n*(n+2))*(2*x^2+(n^2+n-4)*x-(n-1)*(n+2))*x^(-(3*n+2)/n),y[x],x,IncludeSingularSolutions -> True]
Not solved