Internal problem ID [10818]
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. Equations
of the form \((g_1(x)+g_0(x))y'=f_2(x) y^2+f_1(x) y+f_0(x)\)
Problem number: 1.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class C`], _exact, _rational, [_Abel, `2nd type`, `class A`]]
\[ \boxed {\left (A y+B x +a \right ) y^{\prime }+B y=-k x -b} \]
✓ Solution by Maple
Time used: 0.344 (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)
\[ y \left (x \right ) = \frac {-\sqrt {-\left (A k -B^{2}\right ) \left (\left (k x +b \right ) A -B^{2} x -B a \right )^{2} c_{1}^{2}+A}+\left (k \left (-B x -a \right ) A +x \,B^{3}+a \,B^{2}\right ) c_{1}}{A c_{1} \left (A k -B^{2}\right )} \]
✓ Solution by Mathematica
Time used: 18.19 (sec). Leaf size: 106
DSolve[(A*y[x]+B*x+a)*y'[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*}