Internal problem ID [9986]
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: 4.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {y y^{\prime }-\frac {y}{\sqrt {a x +b}}-1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.132 (sec). Leaf size: 171
dsolve(y(x)*diff(y(x),x)=(a*x+b)^(-1/2)*y(x)+1,y(x), singsol=all)
\[ -\frac {2 \arctanh \left (\frac {\sqrt {x a +b}\, y \relax (x ) a -x a -b}{\sqrt {\left (2 a +1\right ) \left (x a +b \right )^{2}}}\right ) x a}{\sqrt {\left (2 a +1\right ) \left (x a +b \right )^{2}}}+\ln \left (a y \relax (x )^{2} \sqrt {x a +b}-2 \sqrt {x a +b}\, x a -2 y \relax (x ) a x -2 \sqrt {x a +b}\, b -2 b y \relax (x )\right )-\frac {2 \arctanh \left (\frac {\sqrt {x a +b}\, y \relax (x ) a -x a -b}{\sqrt {\left (2 a +1\right ) \left (x a +b \right )^{2}}}\right ) b}{\sqrt {\left (2 a +1\right ) \left (x a +b \right )^{2}}}-\frac {\ln \left (x a +b \right )}{2}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.154 (sec). Leaf size: 90
DSolve[y[x]*y'[x]==(a*x+b)^(-1/2)*y[x]+1,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {\frac {2 \text {ArcTan}\left (\frac {\frac {a y(x)}{\sqrt {a x+b}}-1}{\sqrt {-2 a-1}}\right )}{\sqrt {-2 a-1}}+\log \left (-\frac {a y(x)^2}{a x+b}+\frac {2 y(x)}{\sqrt {a x+b}}+2\right )}{a}=\frac {\log (a x+b)}{a}+c_1,y(x)\right ] \]