61.22.36 problem 36

Internal problem ID [12361]
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. subsection 1.3.1-2. Solvable equations and their solutions
Problem number : 36
Date solved : Tuesday, January 28, 2025 at 07:56:55 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y^{\prime } y-y&=A \sqrt {x}+2 A^{2}+\frac {B}{\sqrt {x}} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 407

dsolve(y(x)*diff(y(x),x)-y(x)=A*x^(1/2)+2*A^2+B*x^(-1/2),y(x), singsol=all)
 

\[ \frac {-c_{1} \left (\sqrt {\frac {A^{3}-B}{A^{3}}}\, A -A -\sqrt {x}\right ) \operatorname {BesselI}\left (\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}-y A +x A +B}{A^{3}}}\right )+A \operatorname {BesselI}\left (1+\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}-y A +x A +B}{A^{3}}}\right ) \sqrt {\frac {2 A^{2} \sqrt {x}-y A +x A +B}{A^{3}}}\, c_{1} +\operatorname {BesselK}\left (1+\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}-y A +x A +B}{A^{3}}}\right ) \sqrt {\frac {2 A^{2} \sqrt {x}-y A +x A +B}{A^{3}}}\, A +\left (\sqrt {\frac {A^{3}-B}{A^{3}}}\, A -A -\sqrt {x}\right ) \operatorname {BesselK}\left (\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}-y A +x A +B}{A^{3}}}\right )}{\left (-\sqrt {\frac {A^{3}-B}{A^{3}}}\, A +A +\sqrt {x}\right ) \operatorname {BesselI}\left (\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}-y A +x A +B}{A^{3}}}\right )+\operatorname {BesselI}\left (1+\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}-y A +x A +B}{A^{3}}}\right ) \sqrt {\frac {2 A^{2} \sqrt {x}-y A +x A +B}{A^{3}}}\, A} = 0 \]

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

DSolve[y[x]*D[y[x],x]-y[x]==A*x^(1/2)+2*A^2+B*x^(-1/2),y[x],x,IncludeSingularSolutions -> True]
 

Not solved