Internal problem ID [10787]
Internal file name [OUTPUT/9735_Monday_June_06_2022_04_48_29_PM_62088419/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: 51.
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.
\[ \boxed {y y^{\prime }-\frac {a \left (x +4\right ) y}{5 x^{\frac {8}{5}}}=\frac {a^{2} \left (x -1\right ) \left (3 x +7\right )}{5 x^{\frac {11}{5}}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 5 y y^{\prime } x^{\frac {11}{5}}-a y x^{\frac {8}{5}}-4 a y x^{\frac {3}{5}}-3 a^{2} x^{2}-4 a^{2} x +7 a^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a y x^{\frac {8}{5}}+4 a y x^{\frac {3}{5}}+3 a^{2} x^{2}+4 a^{2} x -7 a^{2}}{5 y x^{\frac {11}{5}}} \end {array} \]
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: 0.015 (sec). Leaf size: 190
dsolve(y(x)*diff(y(x),x)-1/5*a*(x+4)*x^(-8/5)*y(x)=1/5*a^2*(x-1)*(3*x+7)*x^(-11/5),y(x), singsol=all)
\[ \frac {\frac {360 \,2^{\frac {1}{3}} \sqrt {17}\, \sqrt {-\frac {\left (-1+x \right ) a +y \left (x \right ) x^{\frac {3}{5}}}{x^{\frac {3}{5}} \left (y \left (x \right )+a \,x^{\frac {2}{5}}\right )}}\, 91^{\frac {5}{6}} \left (x -\frac {21}{4}\right ) \left (\frac {\left (3 x +7\right ) a +3 y \left (x \right ) x^{\frac {3}{5}}}{x^{\frac {3}{5}} \left (y \left (x \right )+a \,x^{\frac {2}{5}}\right )}\right )^{\frac {7}{6}}}{4444531}+31255875 x \left (\int _{}^{-\frac {315 \left (4 y \left (x \right ) x^{\frac {3}{5}}+4 a x -21 a \right )}{884 \left (y \left (x \right ) x^{\frac {3}{5}}+a x \right )}}\frac {\sqrt {52 \textit {\_a} -315}\, \left (68 \textit {\_a} +315\right )^{\frac {1}{6}} \textit {\_a}}{\left (11492 \textit {\_a}^{2}-53235 \textit {\_a} -99225\right ) \left (221 \textit {\_a} +315\right )^{\frac {5}{3}}}d \textit {\_a} +\frac {c_{1}}{31255875}\right ) \left (\frac {a}{x^{\frac {3}{5}} \left (y \left (x \right )+a \,x^{\frac {2}{5}}\right )}\right )^{\frac {5}{3}}}{x \left (\frac {a}{x^{\frac {3}{5}} \left (y \left (x \right )+a \,x^{\frac {2}{5}}\right )}\right )^{\frac {5}{3}}} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-1/5*a*(x+4)*x^(-8/5)*y[x]==1/5*a^2*(x-1)*(3*x+7)*x^(-11/5),y[x],x,IncludeSingularSolutions -> True]
Timed out