61.23.5 problem 5

Internal problem ID [12406]
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 1.3.2.
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 07:58:10 PM
CAS classification : [[_Abel, `2nd type`, `class B`]]

\begin{align*} y^{\prime } y&=\frac {3 y}{\sqrt {a \,x^{{3}/{2}}+8 x}}+1 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 293

dsolve(y(x)*diff(y(x),x)=3*(a*x^(3/2)+8*x)^(-1/2)*y(x)+1,y(x), singsol=all)
 

\[ \frac {{\left (-\frac {a \sqrt {x}\, \left (-2 a \,x^{{3}/{2}}+\sqrt {x}\, a y^{2}-8 \sqrt {x \left (8+a \sqrt {x}\right )}\, y-16 x \right )}{\left (\sqrt {x}\, a y-4 \sqrt {x \left (8+a \sqrt {x}\right )}\right )^{2}}\right )}^{{1}/{4}} \sqrt {2 a \sqrt {x}+16}\, a \sqrt {x}\, y+4 \left (\sqrt {x}\, a y-4 \sqrt {x \left (8+a \sqrt {x}\right )}\right ) \left (\int _{}^{-\frac {\sqrt {2 a \sqrt {x}+16}\, \sqrt {x \left (8+a \sqrt {x}\right )}}{\sqrt {x}\, a y-4 \sqrt {x \left (8+a \sqrt {x}\right )}}}\frac {\left (\textit {\_a}^{2}-1\right )^{{1}/{4}}}{\sqrt {\textit {\_a}}}d \textit {\_a} +\frac {c_{1}}{4}\right ) \sqrt {-\frac {\sqrt {2 a \sqrt {x}+16}\, \sqrt {x \left (8+a \sqrt {x}\right )}}{\sqrt {x}\, a y-4 \sqrt {x \left (8+a \sqrt {x}\right )}}}}{\sqrt {-\frac {\sqrt {2 a \sqrt {x}+16}\, \sqrt {x \left (8+a \sqrt {x}\right )}}{\sqrt {x}\, a y-4 \sqrt {x \left (8+a \sqrt {x}\right )}}}\, \left (\sqrt {x}\, a y-4 \sqrt {x \left (8+a \sqrt {x}\right )}\right )} = 0 \]

✗ Solution by Mathematica

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

DSolve[y[x]*D[y[x],x]==3*(a*x^(3/2)+8*x)^(-1/2)*y[x]+1,y[x],x,IncludeSingularSolutions -> True]
 

Not solved