Internal problem ID [9937]
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`]]
\[ \boxed {y^{\prime } y-y-A \left (n +2\right ) \left (\sqrt {x}+2 \left (n +2\right ) A +\frac {\left (3+2 n \right ) A^{2}}{\sqrt {x}}\right )=0} \]
✓ Solution by Maple
Time used: 0.0 (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)
\[ c_{1} +\frac {\left (A \sqrt {\frac {\left (1+n \right )^{2}}{\left (n +2\right )^{2}}}\, \left (n +2\right )-\sqrt {x}+\left (-n -2\right ) A \right ) \operatorname {BesselK}\left (\sqrt {\frac {\left (1+n \right )^{2}}{\left (n +2\right )^{2}}}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (3+2 n \right ) A^{2}+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}\right )+\operatorname {BesselK}\left (1+\sqrt {\frac {\left (1+n \right )^{2}}{\left (n +2\right )^{2}}}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (3+2 n \right ) A^{2}+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}\right ) \sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (3+2 n \right ) A^{2}+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}\, A \left (n +2\right )}{\left (-A \sqrt {\frac {\left (1+n \right )^{2}}{\left (n +2\right )^{2}}}\, \left (n +2\right )+\sqrt {x}+\left (n +2\right ) A \right ) \operatorname {BesselI}\left (\sqrt {\frac {\left (1+n \right )^{2}}{\left (n +2\right )^{2}}}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (3+2 n \right ) A^{2}+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}\right )+A \operatorname {BesselI}\left (1+\sqrt {\frac {\left (1+n \right )^{2}}{\left (n +2\right )^{2}}}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (3+2 n \right ) A^{2}+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}\right ) \sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (3+2 n \right ) A^{2}+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}\, \left (n +2\right )} = 0 \]
✗ 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+(2*n+3)*A^2*x^(-1/2)),y[x],x,IncludeSingularSolutions -> True]
Not solved