Internal problem ID [9942]
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.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {y^{\prime } y-y-A \sqrt {x}-2 A^{2}-\frac {B}{\sqrt {x}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 273
dsolve(y(x)*diff(y(x),x)-y(x)=A*x^(1/2)+2*A^2+B*x^(-1/2),y(x), singsol=all)
\[ c_{1}+\frac {\left (\sqrt {\frac {A^{3}-B}{A^{3}}}\, A -A -\sqrt {x}\right ) \BesselK \left (\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}+\left (x -y \relax (x )\right ) A +B}{A^{3}}}\right )+\BesselK \left (1+\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}+\left (x -y \relax (x )\right ) A +B}{A^{3}}}\right ) \sqrt {\frac {2 A^{2} \sqrt {x}+\left (x -y \relax (x )\right ) A +B}{A^{3}}}\, A}{\left (-\sqrt {\frac {A^{3}-B}{A^{3}}}\, A +A +\sqrt {x}\right ) \BesselI \left (\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}+\left (x -y \relax (x )\right ) A +B}{A^{3}}}\right )+A \sqrt {\frac {2 A^{2} \sqrt {x}+\left (x -y \relax (x )\right ) A +B}{A^{3}}}\, \BesselI \left (1+\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}+\left (x -y \relax (x )\right ) A +B}{A^{3}}}\right )} = 0 \]
✗ 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