Internal problem ID [10702]
Internal file name [OUTPUT/9650_Monday_June_06_2022_03_17_50_PM_13823638/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: 54.
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.
\[ \boxed {y y^{\prime }-y=6 x +\frac {A}{x^{4}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -y y^{\prime } x^{4}+y x^{4}+6 x^{5}+A =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-y x^{4}-6 x^{5}-A}{y x^{4}} \end {array} \]
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 found: 2 potential symmetries. Proceeding with integration step <- Abel successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 217
dsolve(y(x)*diff(y(x),x)-y(x)=6*x+A*x^(-4),y(x), singsol=all)
\[ c_{1} +\frac {5 \,5^{\frac {2}{3}} \left (x +\frac {y \left (x \right )}{2}\right ) \left (x A 3^{\frac {5}{6}} \operatorname {hypergeom}\left (\left [\frac {1}{6}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -\frac {2 A}{3 x^{3} \left (2 x +y \left (x \right )\right )^{2}}\right ) \left (\frac {12 x^{5}+12 y \left (x \right ) x^{4}+3 y \left (x \right )^{2} x^{3}+2 A}{x^{9} \left (2 x +y \left (x \right )\right )^{6}}\right )^{\frac {1}{6}}+\frac {24 \left (\frac {\left (-\frac {1}{x^{\frac {3}{2}} \left (2 x +y \left (x \right )\right )}\right )^{\frac {2}{3}} y \left (x \right )}{6}+\left (-\frac {1}{x^{\frac {3}{2}} \left (2 x +y \left (x \right )\right )}\right )^{\frac {5}{3}} x^{\frac {5}{2}} \left (x +\frac {y \left (x \right )}{2}\right )\right ) \left (6 x^{5}+6 y \left (x \right ) x^{4}+\frac {3 y \left (x \right )^{2} x^{3}}{2}+A \right )}{5}\right )}{2 \left (-\frac {1}{x^{\frac {3}{2}} \left (2 x +y \left (x \right )\right )}\right )^{\frac {7}{3}} x^{\frac {11}{2}} \left (\frac {12 x^{5}+12 y \left (x \right ) x^{4}+3 y \left (x \right )^{2} x^{3}+2 A}{x^{3} \left (2 x +y \left (x \right )\right )^{2}}\right )^{\frac {1}{6}} \left (2 x +y \left (x \right )\right )^{4}} = 0 \]
✓ Solution by Mathematica
Time used: 2.079 (sec). Leaf size: 213
DSolve[y[x]*y'[x]-y[x]==6*x+A*x^(-4),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [c_1=\frac {i \left (-\frac {2 A+12 x^5+12 x^4 y(x)+3 x^3 y(x)^2}{A}\right )^{5/6} \left (-10\ 2^{5/6} x^5 \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {3}{2},-\frac {3 x^3 (2 x+y(x))^2}{2 A}\right )-5\ 2^{5/6} x^4 y(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {3}{2},-\frac {3 x^3 (2 x+y(x))^2}{2 A}\right )+A \left (\frac {2 A+12 x^5+12 x^4 y(x)+3 x^3 y(x)^2}{A}\right )^{5/6}\right )}{2 \sqrt [3]{2} \sqrt {3} \sqrt {A} x^{5/2} \left (\frac {2 A+12 x^5+12 x^4 y(x)+3 x^3 y(x)^2}{A}\right )^{5/6}},y(x)\right ] \]