61.25.1 problem 1

Internal problem ID [12494]
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 1.3.4-2.
Problem number : 1
Date solved : Tuesday, January 28, 2025 at 03:17:49 AM
CAS classification : [[_homogeneous, `class C`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} \left (A y+B x +a \right ) y^{\prime }+B y+k x +b&=0 \end{align*}

✓ Solution by Maple

Time used: 1.197 (sec). Leaf size: 87

dsolve((A*y(x)+B*x+a)*diff(y(x),x)+B*y(x)+k*x+b=0,y(x), singsol=all)
 

\[ y = \frac {-\sqrt {-\left (A k -B^{2}\right ) \left (\left (k x +b \right ) A -B^{2} x -a B \right )^{2} c_{1}^{2}+A}+\left (k \left (-B x -a \right ) A +x \,B^{3}+B^{2} a \right ) c_{1}}{c_{1} A \left (A k -B^{2}\right )} \]

✓ Solution by Mathematica

Time used: 17.029 (sec). Leaf size: 106

DSolve[(A*y[x]+B*x+a)*D[y[x],x]+B*y[x]+k*x+b==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {\frac {\sqrt {\frac {(a+B x)^2}{A}+A c_1-x (2 b+k x)}}{\sqrt {\frac {1}{A}}}+a+B x}{A} \\ y(x)\to -\frac {a+B x}{A}+\sqrt {\frac {1}{A}} \sqrt {\frac {(a+B x)^2}{A}+A c_1-x (2 b+k x)} \\ \end{align*}