Internal problem ID [10755]
Internal file name [OUTPUT/9703_Monday_June_06_2022_03_29_28_PM_56850454/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: 19.
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.
\[ \boxed {y y^{\prime }+\frac {a \left (6 x -1\right ) y}{2 x}=-\frac {a^{2} \left (x -1\right ) \left (4 x -1\right )}{2 x}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 a^{2} x^{2}+2 y y^{\prime } x +6 a x y-5 a^{2} x -a y+a^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-4 a^{2} x^{2}-6 a x y+5 a^{2} x +a y-a^{2}}{2 x y} \end {array} \]
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: 0.015 (sec). Leaf size: 364
dsolve(y(x)*diff(y(x),x)+1/2*a*(6*x-1)*1/x*y(x)=-1/2*a^2*(x-1)*(4*x-1)*1/x,y(x), singsol=all)
\[ c_{1} +\frac {\sqrt {2}\, \left (\frac {i \left (i \sqrt {-x}\, a +2 a x +y \left (x \right )-a \right ) \sqrt {-x}}{x a}\right )^{\frac {3}{2}} \left (-\frac {i \left (i \sqrt {-x}\, a +2 a x +y \left (x \right )-a \right ) \sqrt {-x}\, \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {3}{2}\right ], \left [\frac {7}{2}\right ], \frac {i \left (i \sqrt {-x}\, a +2 a x +y \left (x \right )-a \right ) \sqrt {-x}}{2 x a}\right )}{8 x a}+\frac {5 \left (4 i \sqrt {2}\, x -i \sqrt {2}-6 i \sqrt {-x}+4 x +2\right ) \operatorname {hypergeom}\left (\left [-\frac {1}{2}, \frac {1}{2}\right ], \left [\frac {5}{2}\right ], \frac {i \left (i \sqrt {-x}\, a +2 a x +y \left (x \right )-a \right ) \sqrt {-x}}{2 x a}\right )}{4 \left (4 i \sqrt {2}\, x +i \sqrt {2}-2 i \sqrt {-x}-4 \sqrt {2}\, \sqrt {-x}+4 x -2\right )}\right )}{2 \left (\frac {3}{2}-\frac {4 i \sqrt {2}\, x -i \sqrt {2}-6 i \sqrt {-x}+4 x +2}{2 \left (4 i \sqrt {2}\, x +i \sqrt {2}-2 i \sqrt {-x}-4 \sqrt {2}\, \sqrt {-x}+4 x -2\right )}\right ) \operatorname {hypergeom}\left (\left [-2, -1\right ], \left [-\frac {1}{2}\right ], \frac {i \left (i \sqrt {-x}\, a +2 a x +y \left (x \right )-a \right ) \sqrt {-x}}{2 x a}\right )+\frac {4 i \left (i \sqrt {-x}\, a +2 a x +y \left (x \right )-a \right ) \sqrt {-x}}{x a}} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]+1/2*a*(6*x-1)*1/x*y[x]==-1/2*a^2*(x-1)*(4*x-1)*1/x,y[x],x,IncludeSingularSolutions -> True]
Not solved