Internal problem ID [10789]
Internal file name [OUTPUT/9737_Monday_June_06_2022_04_48_34_PM_47478066/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: 53.
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.
\[ \boxed {y y^{\prime }+\frac {a \left (x -6\right ) y}{5 x^{\frac {7}{5}}}=\frac {2 a^{2} \left (x -1\right ) \left (x +4\right )}{5 x^{\frac {9}{5}}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 5 y y^{\prime } x^{\frac {9}{5}}+a y x^{\frac {7}{5}}-6 a y x^{\frac {2}{5}}-2 a^{2} x^{2}-6 a^{2} x +8 a^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-a y x^{\frac {7}{5}}+6 a y x^{\frac {2}{5}}+2 a^{2} x^{2}+6 a^{2} x -8 a^{2}}{5 x^{\frac {9}{5}} y} \end {array} \]
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 <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 156
dsolve(y(x)*diff(y(x),x)+1/5*a*(x-6)*x^(-7/5)*y(x)=2/5*a^2*(x-1)*(x+4)*x^(-9/5),y(x), singsol=all)
\[ c_{1} -\frac {80 \sqrt {3}\, \left (a y \left (x \right ) x^{\frac {2}{5}}+\frac {x^{\frac {4}{5}} y \left (x \right )^{2}}{8}+\frac {\left (y \left (x \right ) x^{\frac {7}{5}}-2 a \left (x +24\right ) \left (-1+x \right )\right ) a}{24}\right ) \left (a y \left (x \right ) x^{\frac {2}{5}}+\frac {x^{\frac {4}{5}} y \left (x \right )^{2}}{8}+\frac {a \left (y \left (x \right ) x^{\frac {7}{5}}+\frac {a \left (x +4\right )^{2}}{2}\right )}{4}\right ) \sqrt {\frac {-y \left (x \right ) x^{\frac {2}{5}}-a x +a}{x^{\frac {2}{5}} \left (y \left (x \right )+x^{\frac {3}{5}} a \right )}}}{9 \left (\frac {a}{x^{\frac {2}{5}} \left (y \left (x \right )+x^{\frac {3}{5}} a \right )}\right )^{\frac {5}{2}} \left (a \left (x +4\right )+y \left (x \right ) x^{\frac {2}{5}}\right )^{2} \left (y \left (x \right ) x^{\frac {2}{5}}+a x \right )^{2} x} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]+1/5*a*(x-6)*x^(-7/5)*y[x]==2/5*a^2*(x-1)*(x+4)*x^(-9/5),y[x],x,IncludeSingularSolutions -> True]
Timed out