Internal problem ID [9987]
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. Equations of the
form \(y y'=f(x) y+1\)
Problem number: 5.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {y^{\prime } y-\frac {3 y}{\sqrt {a \,x^{\frac {3}{2}}+8 x}}-1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 291
dsolve(y(x)*diff(y(x),x)=3*(a*x^(3/2)+8*x)^(-1/2)*y(x)+1,y(x), singsol=all)
\[ c_{1}+\frac {4 \left (\int _{}^{-\frac {\sqrt {2 a \sqrt {x}+16}\, \sqrt {x \left (8+a \sqrt {x}\right )}}{\sqrt {x}\, a y \relax (x )-4 \sqrt {x \left (8+a \sqrt {x}\right )}}}\frac {\left (\textit {\_a}^{2}-1\right )^{\frac {1}{4}}}{\sqrt {\textit {\_a}}}d \textit {\_a} \right ) \sqrt {-\frac {\sqrt {2 a \sqrt {x}+16}\, \sqrt {x \left (8+a \sqrt {x}\right )}}{\sqrt {x}\, a y \relax (x )-4 \sqrt {x \left (8+a \sqrt {x}\right )}}}\, \left (\sqrt {x}\, a y \relax (x )-4 \sqrt {x \left (8+a \sqrt {x}\right )}\right )+\left (-\frac {a \sqrt {x}\, \left (-2 a \,x^{\frac {3}{2}}+\sqrt {x}\, a y \relax (x )^{2}-8 \sqrt {x \left (8+a \sqrt {x}\right )}\, y \relax (x )-16 x \right )}{\left (\sqrt {x}\, a y \relax (x )-4 \sqrt {x \left (8+a \sqrt {x}\right )}\right )^{2}}\right )^{\frac {1}{4}} \sqrt {2 a \sqrt {x}+16}\, a \sqrt {x}\, y \relax (x )}{\sqrt {-\frac {\sqrt {2 a \sqrt {x}+16}\, \sqrt {x \left (8+a \sqrt {x}\right )}}{\sqrt {x}\, a y \relax (x )-4 \sqrt {x \left (8+a \sqrt {x}\right )}}}\, \left (\sqrt {x}\, a y \relax (x )-4 \sqrt {x \left (8+a \sqrt {x}\right )}\right )} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]==3*(a*x^(3/2)+8*x)^(-1/2)*y[x]+1,y[x],x,IncludeSingularSolutions -> True]
Not solved