61.12.7 problem 44

Internal problem ID [12218]
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
Date solved : Tuesday, January 28, 2025 at 07:35:25 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} y^{\prime }&=a \cot \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 41

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)
 

\[ y = b \,x^{n}+c +\frac {1}{c_{1} -a \left (\int \left (\frac {-1+\cot \left (\mu \right ) \cot \left (\lambda x \right )}{\cot \left (\mu \right )+\cot \left (\lambda x \right )}\right )^{k}d x \right )} \]

✓ Solution by Mathematica

Time used: 1.470 (sec). Leaf size: 74

DSolve[D[y[x],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) \operatorname {Hypergeometric2F1}\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*}