Internal problem ID [10685]
Internal file name [OUTPUT/9633_Monday_June_06_2022_03_16_58_PM_33792140/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: 37.
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 A`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-y=2 A^{2}-A \sqrt {x}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-y=2 A^{2}-A \sqrt {x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y+2 A^{2}-A \sqrt {x}}{y} \end {array} \]
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 Looking for potential symmetries Looking for potential symmetries <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 228
dsolve(y(x)*diff(y(x),x)-y(x)=2*A^2-A*x^(1/2),y(x), singsol=all)
\[ \frac {\left (-2 A +\sqrt {x}\right ) \operatorname {BesselK}\left (1, -\sqrt {-\frac {2 A \sqrt {x}-x +y \left (x \right )}{A^{2}}}\right )+\operatorname {BesselK}\left (0, -\sqrt {-\frac {2 A \sqrt {x}-x +y \left (x \right )}{A^{2}}}\right ) \sqrt {-\frac {2 A \sqrt {x}-x +y \left (x \right )}{A^{2}}}\, A +c_{1} \left (A \operatorname {BesselI}\left (0, \sqrt {-\frac {2 A \sqrt {x}-x +y \left (x \right )}{A^{2}}}\right ) \sqrt {-\frac {2 A \sqrt {x}-x +y \left (x \right )}{A^{2}}}+\left (-2 A +\sqrt {x}\right ) \operatorname {BesselI}\left (1, \sqrt {-\frac {2 A \sqrt {x}-x +y \left (x \right )}{A^{2}}}\right )\right )}{A \operatorname {BesselI}\left (0, \sqrt {-\frac {2 A \sqrt {x}-x +y \left (x \right )}{A^{2}}}\right ) \sqrt {-\frac {2 A \sqrt {x}-x +y \left (x \right )}{A^{2}}}+\left (-2 A +\sqrt {x}\right ) \operatorname {BesselI}\left (1, \sqrt {-\frac {2 A \sqrt {x}-x +y \left (x \right )}{A^{2}}}\right )} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==2*A^2-A*x^(1/2),y[x],x,IncludeSingularSolutions -> True]
Not solved