61.23.4 problem 4

Internal problem ID [12405]
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 : 4
Date solved : Tuesday, January 28, 2025 at 02:58:37 AM
CAS classification : [[_1st_order, _with_linear_symmetries], [_Abel, `2nd type`, `class B`]]

\begin{align*} y^{\prime } y&=\frac {y}{\sqrt {a x +b}}+1 \end{align*}

✓ Solution by Maple

Time used: 0.437 (sec). Leaf size: 153

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

\[ \frac {2 \,\operatorname {arctanh}\left (\frac {-\sqrt {a x +b}\, y a +a x +b}{\sqrt {\left (2 a +1\right ) \left (a x +b \right )^{2}}}\right ) a x}{\sqrt {\left (2 a +1\right ) \left (a x +b \right )^{2}}}+\ln \left (\left (a y^{2}-2 a x -2 b \right ) \sqrt {a x +b}-2 \left (a x +b \right ) y\right )+\frac {2 \,\operatorname {arctanh}\left (\frac {-\sqrt {a x +b}\, y a +a x +b}{\sqrt {\left (2 a +1\right ) \left (a x +b \right )^{2}}}\right ) b}{\sqrt {\left (2 a +1\right ) \left (a x +b \right )^{2}}}-\frac {\ln \left (a x +b \right )}{2}-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.072 (sec). Leaf size: 50

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

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{\sqrt {b+a x}}}\frac {1}{-\frac {1}{2} a K[1]+1+\frac {1}{K[1]}}dK[1]=\frac {\log (a x+b)}{a}+c_1,y(x)\right ] \]