Internal problem ID [10768]
Internal file name [OUTPUT/9716_Monday_June_06_2022_04_47_32_PM_87061287/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: 32.
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.
\[ \boxed {y y^{\prime }-\frac {a \left (x +1\right ) y}{2 x^{\frac {7}{4}}}=\frac {a^{2} \left (x -1\right ) \left (x +5\right )}{4 x^{\frac {5}{2}}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 y y^{\prime } x^{\frac {17}{4}}-x^{\frac {15}{4}} a^{2}-4 x^{\frac {11}{4}} a^{2}+5 a^{2} x^{\frac {7}{4}}-2 a y x^{\frac {7}{2}}-2 x^{\frac {5}{2}} y a =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x^{\frac {15}{4}} a^{2}+4 x^{\frac {11}{4}} a^{2}-5 a^{2} x^{\frac {7}{4}}+2 a y x^{\frac {7}{2}}+2 x^{\frac {5}{2}} y a}{4 y x^{\frac {17}{4}}} \end {array} \]
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)-(1/2)*y(x)*(x^2+12*x-25)/(x*(x+5)*(x^(1/4)-1)*(x^(1/4)+1)*(x^(1/2)+1)), y(x)` 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)-1/2*a*(x+1)*x^(-7/4)*y(x)=1/4*a^2*(x-1)*(x+5)*x^(-5/2),y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-1/2*a*(x+1)*x^(-7/4)*y[x]==1/4*a^2*(x-1)*(x+5)*x^(-5/2),y[x],x,IncludeSingularSolutions -> True]
Timed out