Internal problem ID [10652]
Internal file name [OUTPUT/9600_Monday_June_06_2022_03_13_15_PM_16949097/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: 4.
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.
\[ \boxed {y y^{\prime }-y=2 A \left (\sqrt {x}+4 A +\frac {3 A^{2}}{\sqrt {x}}\right )} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 6 A^{3}+8 A^{2} \sqrt {x}+2 A x -y y^{\prime } \sqrt {x}+y \sqrt {x}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-6 A^{3}-8 A^{2} \sqrt {x}-2 A x -y \sqrt {x}}{y \sqrt {x}} \end {array} \]
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: 0.015 (sec). Leaf size: 120
dsolve(y(x)*diff(y(x),x)-y(x)=2*A*(x^(1/2)+4*A+3*A^2*x^(-1/2)),y(x), singsol=all)
\[ \frac {-\sqrt {\frac {-6 A^{2}-8 A \sqrt {x}-2 x +2 y \left (x \right )}{y \left (x \right )}}\, \sqrt {2}+4 \sqrt {-\frac {A^{2}}{y \left (x \right )}}\, \operatorname {arctanh}\left (\frac {\sqrt {-\frac {A^{2}}{y \left (x \right )}}\, \left (3 A +\sqrt {x}\right )}{\sqrt {\frac {-3 A^{2}-4 A \sqrt {x}-x +y \left (x \right )}{y \left (x \right )}}\, A}\right )+c_{1} \sqrt {-\frac {A^{2}}{y \left (x \right )}}}{\sqrt {-\frac {A^{2}}{y \left (x \right )}}} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==2*A*(x^(1/2)+4*A+3*A^2*x^(-1/2)),y[x],x,IncludeSingularSolutions -> True]
Not solved