Internal problem ID [10695]
Internal file name [OUTPUT/9643_Monday_June_06_2022_03_17_27_PM_31172118/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. Form \(y y'-y=f(x)\). subsection 1.3.1-2.
Solvable equations and their solutions
Problem number: 47.
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.
\[ \boxed {y y^{\prime }-y=12 x +\frac {A}{x^{\frac {5}{2}}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -y y^{\prime } x^{\frac {5}{2}}+y x^{\frac {5}{2}}+12 x^{\frac {7}{2}}+A =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-y x^{\frac {5}{2}}-12 x^{\frac {7}{2}}-A}{y x^{\frac {5}{2}}} \end {array} \]
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: 110
dsolve(y(x)*diff(y(x),x)-y(x)=12*x+A*x^(-5/2),y(x), singsol=all)
\[ c_{1} +\frac {12 \sqrt {3}\, \left (2^{\frac {2}{3}} \left (\frac {3 y \left (x \right )^{2} x^{\frac {3}{2}}}{4}-6 y \left (x \right ) x^{\frac {5}{2}}+A +12 x^{\frac {7}{2}}\right ) \left (\frac {48 x^{\frac {7}{2}}-24 y \left (x \right ) x^{\frac {5}{2}}+3 y \left (x \right )^{2} x^{\frac {3}{2}}+4 A}{A}\right )^{\frac {1}{6}}-56 \operatorname {hypergeom}\left (\left [-\frac {1}{6}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -\frac {3 x^{\frac {3}{2}} \left (-4 x +y \left (x \right )\right )^{2}}{4 A}\right ) \left (x -\frac {y \left (x \right )}{4}\right ) x^{\frac {5}{2}}\right )}{\sqrt {-A \,x^{\frac {7}{2}}}} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==12*x+A*x^(-5/2),y[x],x,IncludeSingularSolutions -> True]
Not solved