Internal problem ID [10801]
Internal file name [OUTPUT/9749_Wednesday_June_08_2022_05_53_31_PM_98114476/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: 65.
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.
\[ \boxed {y y^{\prime }+\frac {a \left (\frac {\left (3 n +5\right ) x}{2}+\frac {n -1}{n +1}\right ) x^{-\frac {n +4}{n +3}} y}{n +3}=-\frac {a^{2} \left (\left (n +1\right ) x^{2}-\frac {\left (n^{2}+2 n +5\right ) x}{n +1}+\frac {4}{n +1}\right ) x^{-\frac {5+n}{n +3}}}{2 n +6}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{-\frac {5+n}{n +3}} a^{2} n^{2} x^{2}+3 y x^{-\frac {n +4}{n +3}} a \,n^{2} x -x^{-\frac {5+n}{n +3}} a^{2} n^{2} x +2 x^{-\frac {5+n}{n +3}} a^{2} n \,x^{2}+8 y x^{-\frac {n +4}{n +3}} a n x -2 x^{-\frac {5+n}{n +3}} a^{2} n x +x^{-\frac {5+n}{n +3}} a^{2} x^{2}+2 y y^{\prime } n^{2}+2 y x^{-\frac {n +4}{n +3}} a n +5 y x^{-\frac {n +4}{n +3}} a x -5 x^{-\frac {5+n}{n +3}} a^{2} x +8 y y^{\prime } n -2 a \,x^{-\frac {n +4}{n +3}} y+4 a^{2} x^{-\frac {5+n}{n +3}}+6 y y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-x^{-\frac {5+n}{n +3}} a^{2} n^{2} x^{2}-3 y x^{-\frac {n +4}{n +3}} a \,n^{2} x +x^{-\frac {5+n}{n +3}} a^{2} n^{2} x -2 x^{-\frac {5+n}{n +3}} a^{2} n \,x^{2}-8 y x^{-\frac {n +4}{n +3}} a n x +2 x^{-\frac {5+n}{n +3}} a^{2} n x -x^{-\frac {5+n}{n +3}} a^{2} x^{2}-2 y x^{-\frac {n +4}{n +3}} a n -5 y x^{-\frac {n +4}{n +3}} a x +5 x^{-\frac {5+n}{n +3}} a^{2} x +2 a \,x^{-\frac {n +4}{n +3}} y-4 a^{2} x^{-\frac {5+n}{n +3}}}{2 y n^{2}+8 n y+6 y} \end {array} \]
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)*(3*n^2*x+2*n^2+8*n*x+6*n+5*x-8)/(x*(3+n)*(3*n^2*x+8*n*x+2*n+5*x-2)), y(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)*(n^3*x^2+3*n^2*x^2+2*n^2*x+3*n*x^2+4*n*x+x^2-4*n+10*x-20)/(x*(x-1)*(3+n)*(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 -> 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+3)*((3*n+5)/(2)*x+(n-1)/(n+1))*x^(-(n+4)/(n+3))*y(x)=-a^2/(2*(n+3))*((n+1)*x^2-(n^2+2*n+5)/(n+1)*x+4/(n+1))*x^(-(n+5)/(n+3)),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+3)*((3*n+5)/(2)*x+(n-1)/(n+1))*x^(-(n+4)/(n+3))*y[x]==-a^2/(2*(n+3))*((n+1)*x^2-(n^2+2*n+5)/(n+1)*x+4/(n+1))*x^(-(n+5)/(n+3)),y[x],x,IncludeSingularSolutions -> True]
Timed out