Internal problem ID [10667]
Internal file name [OUTPUT/9615_Monday_June_06_2022_03_15_54_PM_62818123/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: 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.
\[ \boxed {y y^{\prime }-y=2 x +\frac {A}{x^{2}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -y y^{\prime } x^{2}+x^{2} y+2 x^{3}+A =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-x^{2} y-2 x^{3}-A}{x^{2} y} \end {array} \]
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 <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 170
dsolve(y(x)*diff(y(x),x)-y(x)=2*x+A*x^(-2),y(x), singsol=all)
\[ \frac {6 \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {x \left (A^{2}\right )^{\frac {1}{3}}}{A}}\, \left (-2 x +y \left (x \right )\right )}{\sqrt {\frac {\left (4 x^{3}-4 y \left (x \right ) x^{2}+y \left (x \right )^{2} x +2 A \right ) \left (A^{2}\right )^{\frac {1}{3}}}{y \left (x \right )^{2} A}}\, y \left (x \right )}\right ) A +\frac {c_{1}}{6}\right ) x \sqrt {\frac {x \left (A^{2}\right )^{\frac {1}{3}}}{A}}+2 \sqrt {3}\, y \left (x \right ) \left (-x^{3}-\frac {y \left (x \right ) x^{2}}{2}+\frac {y \left (x \right )^{2} x}{2}+A \right ) \sqrt {\frac {\left (4 x^{3}-4 y \left (x \right ) x^{2}+y \left (x \right )^{2} x +2 A \right ) \left (A^{2}\right )^{\frac {1}{3}}}{y \left (x \right )^{2} A}}}{\sqrt {\frac {x \left (A^{2}\right )^{\frac {1}{3}}}{A}}\, x} = 0 \]
✓ Solution by Mathematica
Time used: 2.08 (sec). Leaf size: 233
DSolve[y[x]*y'[x]-y[x]==2*x+A*x^(-2),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [c_1=-\frac {i \sqrt {-\frac {2 A+4 x^3-4 x^2 y(x)+x y(x)^2}{A}} \left (-6 \sqrt {A} x^{3/2} \text {arcsinh}\left (\frac {\sqrt {x} (2 x-y(x))}{\sqrt {2} \sqrt {A}}\right )+x^2 (-y(x)) \sqrt {\frac {2 A+4 x^3-4 x^2 y(x)+x y(x)^2}{A}}+x y(x)^2 \sqrt {\frac {2 A+4 x^3-4 x^2 y(x)+x y(x)^2}{A}}+2 \left (A-x^3\right ) \sqrt {\frac {2 A+4 x^3-4 x^2 y(x)+x y(x)^2}{A}}\right )}{4 \sqrt {A} x^{3/2} \sqrt {\frac {2 A+4 x^3-4 x^2 y(x)+x y(x)^2}{A}}},y(x)\right ] \]