Internal problem ID [10075]
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]]
Solve \begin {gather*} \boxed {\left (A y+B x +a \right ) y^{\prime }+B y+k x +b=0} \end {gather*}
✓ Solution by Maple
Time used: 0.211 (sec). Leaf size: 113
dsolve((A*y(x)+B*x+a)*diff(y(x),x)+B*y(x)+k*x+b=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {-B b +a k +\frac {B \left (x \left (A k -B^{2}\right )+A b -B a \right ) c_{1}+\sqrt {-A c_{1}^{2} k \left (x \left (A k -B^{2}\right )+A b -B a \right )^{2}+B^{2} \left (x \left (A k -B^{2}\right )+A b -B a \right )^{2} c_{1}^{2}+A}}{A c_{1}}}{-A k +B^{2}} \]
✓ Solution by Mathematica
Time used: 0.312 (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*}