Internal problem ID [9940]
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: 34.
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 \left (n +2\right ) \left (\sqrt {x}+2 \left (n +2\right ) A +\frac {\left (n +1\right ) \left (n +3\right ) A^{2}}{\sqrt {x}}\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 309
dsolve(y(x)*diff(y(x),x)-y(x)=A*(n+2)*(x^(1/2)+2*(n+2)*A+(n+1)*(n+3)*A^2*x^(-1/2)),y(x), singsol=all)
\[ c_{1}+\frac {A \sqrt {\frac {2 \left (2+n \right ) A \sqrt {x}+A^{2} \left (n^{2}+4 n +3\right )+x -y \relax (x )}{\left (2+n \right )^{2} A^{2}}}\, \left (2+n \right ) \BesselK \left (\frac {n +3}{2+n}, -\sqrt {\frac {2 \left (2+n \right ) A \sqrt {x}+A^{2} \left (n^{2}+4 n +3\right )+x -y \relax (x )}{\left (2+n \right )^{2} A^{2}}}\right )-\BesselK \left (\frac {1}{2+n}, -\sqrt {\frac {2 \left (2+n \right ) A \sqrt {x}+A^{2} \left (n^{2}+4 n +3\right )+x -y \relax (x )}{\left (2+n \right )^{2} A^{2}}}\right ) \left (\sqrt {x}+\left (n +1\right ) A \right )}{A \sqrt {\frac {2 \left (2+n \right ) A \sqrt {x}+A^{2} \left (n^{2}+4 n +3\right )+x -y \relax (x )}{\left (2+n \right )^{2} A^{2}}}\, \left (2+n \right ) \BesselI \left (\frac {n +3}{2+n}, -\sqrt {\frac {2 \left (2+n \right ) A \sqrt {x}+A^{2} \left (n^{2}+4 n +3\right )+x -y \relax (x )}{\left (2+n \right )^{2} A^{2}}}\right )+\left (\sqrt {x}+\left (n +1\right ) A \right ) \BesselI \left (\frac {1}{2+n}, -\sqrt {\frac {2 \left (2+n \right ) A \sqrt {x}+A^{2} \left (n^{2}+4 n +3\right )+x -y \relax (x )}{\left (2+n \right )^{2} A^{2}}}\right )} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==A*(n+2)*(x^(1/2)+2*(n+2)*A+(n+1)*(n+3)*A^2*x^(-1/2)),y[x],x,IncludeSingularSolutions -> True]
Not solved