Internal problem ID [9941]
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: 35.
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 (2 n +3\right ) A^{2}}{\sqrt {x}}\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 359
dsolve(y(x)*diff(y(x),x)-y(x)=A*(n+2)*(x^(1/2)+2*(n+2)*A+(2*n+3)*A^2*x^(-1/2)),y(x), singsol=all)
\[ c_{1}+\frac {\left (A \sqrt {\frac {\left (n +1\right )^{2}}{\left (2+n \right )^{2}}}\, \left (2+n \right )-\sqrt {x}+\left (-n -2\right ) A \right ) \BesselK \left (\sqrt {\frac {\left (n +1\right )^{2}}{\left (2+n \right )^{2}}}, -\sqrt {\frac {2 \left (2+n \right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y \relax (x )}{\left (2+n \right )^{2} A^{2}}}\right )+\BesselK \left (1+\sqrt {\frac {\left (n +1\right )^{2}}{\left (2+n \right )^{2}}}, -\sqrt {\frac {2 \left (2+n \right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y \relax (x )}{\left (2+n \right )^{2} A^{2}}}\right ) \sqrt {\frac {2 \left (2+n \right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y \relax (x )}{\left (2+n \right )^{2} A^{2}}}\, A \left (2+n \right )}{\left (-A \sqrt {\frac {\left (n +1\right )^{2}}{\left (2+n \right )^{2}}}\, \left (2+n \right )+\sqrt {x}+\left (2+n \right ) A \right ) \BesselI \left (\sqrt {\frac {\left (n +1\right )^{2}}{\left (2+n \right )^{2}}}, -\sqrt {\frac {2 \left (2+n \right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y \relax (x )}{\left (2+n \right )^{2} A^{2}}}\right )+A \sqrt {\frac {2 \left (2+n \right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y \relax (x )}{\left (2+n \right )^{2} A^{2}}}\, \BesselI \left (1+\sqrt {\frac {\left (n +1\right )^{2}}{\left (2+n \right )^{2}}}, -\sqrt {\frac {2 \left (2+n \right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y \relax (x )}{\left (2+n \right )^{2} A^{2}}}\right ) \left (2+n \right )} = 0 \]
✗ 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+(2*n+3)*A^2*x^(-1/2)),y[x],x,IncludeSingularSolutions -> True]
Not solved