Internal problem ID [9761]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-1. Equations with sine
Problem number: 10.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]
\[ \boxed {y^{\prime }-a \sin \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}-b n \,x^{n -1}=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 102
dsolve(diff(y(x),x)=a*sin(lambda*x+mu)^k*(y(x)-b*x^n-c)^2+b*n*x^(n-1),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (-2 a \,x^{n} \left (\sin \left (\lambda x \right ) \cos \left (\mu \right )+\cos \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} b -2 a c \left (\sin \left (\lambda x \right ) \cos \left (\mu \right )+\cos \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k}\right ) \sin \left (\lambda x +\mu \right )^{-k}}{2 a}+\frac {1}{c_{1} -\left (\int a \left (\sin \left (\lambda x \right ) \cos \left (\mu \right )+\cos \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k}d x \right )} \]
✓ Solution by Mathematica
Time used: 3.654 (sec). Leaf size: 93
DSolve[y'[x]==a*Sin[\[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 \sqrt {\cos ^2(\mu +\lambda x)} \sec (\mu +\lambda x) \sin ^{k+1}(\mu +\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\sin ^2(x \lambda +\mu )\right )}{(k+1) \lambda }+c_1}+b x^n+c \\ y(x)\to b x^n+c \\ \end{align*}