Internal problem ID [9799]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-4. Equations with cotangent.
Problem number: 44.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)]], _Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-a \left (\cot ^{k}\left (\lambda x +\mu \right )\right ) \left (y-b \,x^{n}-c \right )^{2}-b n \,x^{n -1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 118
dsolve(diff(y(x),x)=a*cot(lambda*x+mu)^k*(y(x)-b*x^n-c)^2+b*n*x^(n-1),y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (-2 x^{n} \left (\frac {-1+\cot \left (\mu \right ) \cot \left (\lambda x \right )}{\cot \left (\mu \right )+\cot \left (\lambda x \right )}\right )^{k} a b -2 \left (\frac {-1+\cot \left (\mu \right ) \cot \left (\lambda x \right )}{\cot \left (\mu \right )+\cot \left (\lambda x \right )}\right )^{k} a c \right ) \left (\frac {-1+\cot \left (\mu \right ) \cot \left (\lambda x \right )}{\cot \left (\mu \right )+\cot \left (\lambda x \right )}\right )^{-k}}{2 a}+\frac {1}{c_{1}-\left (\int \left (\frac {-1+\cot \left (\mu \right ) \cot \left (\lambda x \right )}{\cot \left (\mu \right )+\cot \left (\lambda x \right )}\right )^{k} a d x \right )} \]
✓ Solution by Mathematica
Time used: 2.5 (sec). Leaf size: 74
DSolve[y'[x]==a*Cot[\[Lambda]*x+mu]^k*(y[x]-b*x^n-c)^2+b*n*x^(n-1),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{\frac {a \cot ^{k+1}(\mu +\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};-\cot ^2(\mu +x \lambda )\right )}{(k+1) \lambda }+c_1}+b x^n+c \\ y(x)\to b x^n+c \\ \end{align*}