Internal problem ID [10078]
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: 4.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational, [_1st_order, _with_symmetry_[F(x),G(x)]], [_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {\left (y+A \,x^{n}+a \right ) y^{\prime }+n A \,x^{n -1} y+k \,x^{m}+b=0} \end {gather*}
✓ Solution by Maple
Time used: 0.013 (sec). Leaf size: 299
dsolve((y(x)+A*x^n+a)*diff(y(x),x)+n*A*x^(n-1)*y(x)+k*x^m+b=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {A m \,x^{n}+A \,x^{n}+a m -\sqrt {A^{2} x^{2 n} m^{2}+2 A^{2} x^{2 n} m +2 A \,x^{n} a \,m^{2}+A^{2} x^{2 n}+4 A \,x^{n} a m +a^{2} m^{2}-2 b \,m^{2} x +2 A \,x^{n} a -2 x^{m +1} k m -2 m^{2} c_{1}+2 m \,a^{2}-4 b x m -2 k \,x^{m +1}-4 c_{1} m +a^{2}-2 b x -2 c_{1}}+a}{m +1} \\ y \relax (x ) = -\frac {A m \,x^{n}+A \,x^{n}+a m +\sqrt {A^{2} x^{2 n} m^{2}+2 A^{2} x^{2 n} m +2 A \,x^{n} a \,m^{2}+A^{2} x^{2 n}+4 A \,x^{n} a m +a^{2} m^{2}-2 b \,m^{2} x +2 A \,x^{n} a -2 x^{m +1} k m -2 m^{2} c_{1}+2 m \,a^{2}-4 b x m -2 k \,x^{m +1}-4 c_{1} m +a^{2}-2 b x -2 c_{1}}+a}{m +1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.472 (sec). Leaf size: 118
DSolve[(y[x]+A*x^n+a)*y'[x]+n*A*x^(n-1)*y[x]+k*x^m+b==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {\frac {1}{x}} \sqrt {x \left (\left (a+A x^n\right )^2-\frac {2 x \left (b m+b+k x^m\right )}{m+1}+c_1\right )}-a-A x^n \\ y(x)\to \sqrt {\frac {1}{x}} \sqrt {x \left (\left (a+A x^n\right )^2-\frac {2 x \left (b m+b+k x^m\right )}{m+1}+c_1\right )}-a-A x^n \\ \end{align*}