Internal problem ID [10684]
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`]]
\[ \boxed {y y^{\prime }-y=A \left (2+n \right ) \left (\sqrt {x}+2 \left (2+n \right ) A +\frac {\left (3+2 n \right ) A^{2}}{\sqrt {x}}\right )} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 474
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)
\[ \frac {-\left (n +2\right ) \left (\operatorname {BesselI}\left (\sqrt {\frac {\left (n +1\right )^{2}}{\left (n +2\right )^{2}}}+1, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}\right ) c_{1} +\operatorname {BesselK}\left (\sqrt {\frac {\left (n +1\right )^{2}}{\left (n +2\right )^{2}}}+1, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}\right )\right ) A \sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}+\left (c_{1} \operatorname {BesselI}\left (\sqrt {\frac {\left (n +1\right )^{2}}{\left (n +2\right )^{2}}}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}\right )-\operatorname {BesselK}\left (\sqrt {\frac {\left (n +1\right )^{2}}{\left (n +2\right )^{2}}}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}\right )\right ) \left (A \sqrt {\frac {\left (n +1\right )^{2}}{\left (n +2\right )^{2}}}\, \left (n +2\right )-\sqrt {x}+\left (-n -2\right ) A \right )}{-A \sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}\, \left (n +2\right ) \operatorname {BesselI}\left (\sqrt {\frac {\left (n +1\right )^{2}}{\left (n +2\right )^{2}}}+1, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}\right )+\operatorname {BesselI}\left (\sqrt {\frac {\left (n +1\right )^{2}}{\left (n +2\right )^{2}}}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (2 n +3\right ) A^{2}+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}\right ) \left (A \sqrt {\frac {\left (n +1\right )^{2}}{\left (n +2\right )^{2}}}\, \left (n +2\right )-\sqrt {x}+\left (-n -2\right ) A \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