Internal problem ID [10684]
Internal file name [OUTPUT/9632_Monday_June_06_2022_03_16_55_PM_1338498/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: 36.
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.
\[ \boxed {y y^{\prime }-y=A \sqrt {x}+2 A^{2}+\frac {B}{\sqrt {x}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 A^{2} \sqrt {x}+A x -y y^{\prime } \sqrt {x}+y \sqrt {x}+B =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-2 A^{2} \sqrt {x}-A x -y \sqrt {x}-B}{y \sqrt {x}} \end {array} \]
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 Looking for potential symmetries Looking for potential symmetries <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 407
dsolve(y(x)*diff(y(x),x)-y(x)=A*x^(1/2)+2*A^2+B*x^(-1/2),y(x), singsol=all)
\[ \frac {-c_{1} \left (\sqrt {\frac {A^{3}-B}{A^{3}}}\, A -A -\sqrt {x}\right ) \operatorname {BesselI}\left (\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}-y \left (x \right ) A +x A +B}{A^{3}}}\right )+A \sqrt {\frac {2 A^{2} \sqrt {x}-y \left (x \right ) A +x A +B}{A^{3}}}\, \operatorname {BesselI}\left (\sqrt {\frac {A^{3}-B}{A^{3}}}+1, -\sqrt {\frac {2 A^{2} \sqrt {x}-y \left (x \right ) A +x A +B}{A^{3}}}\right ) c_{1} +A \sqrt {\frac {2 A^{2} \sqrt {x}-y \left (x \right ) A +x A +B}{A^{3}}}\, \operatorname {BesselK}\left (\sqrt {\frac {A^{3}-B}{A^{3}}}+1, -\sqrt {\frac {2 A^{2} \sqrt {x}-y \left (x \right ) A +x A +B}{A^{3}}}\right )+\operatorname {BesselK}\left (\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}-y \left (x \right ) A +x A +B}{A^{3}}}\right ) \left (\sqrt {\frac {A^{3}-B}{A^{3}}}\, A -A -\sqrt {x}\right )}{A \sqrt {\frac {2 A^{2} \sqrt {x}-y \left (x \right ) A +x A +B}{A^{3}}}\, \operatorname {BesselI}\left (\sqrt {\frac {A^{3}-B}{A^{3}}}+1, -\sqrt {\frac {2 A^{2} \sqrt {x}-y \left (x \right ) A +x A +B}{A^{3}}}\right )+\left (-\sqrt {\frac {A^{3}-B}{A^{3}}}\, A +A +\sqrt {x}\right ) \operatorname {BesselI}\left (\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}-y \left (x \right ) A +x A +B}{A^{3}}}\right )} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==A*x^(1/2)+2*A^2+B*x^(-1/2),y[x],x,IncludeSingularSolutions -> True]
Not solved