Internal problem ID [9778]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-2. Equations with cosine.
Problem number: 23.
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 (\cos ^{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: 0.015 (sec). Leaf size: 106
dsolve(diff(y(x),x)=a*cos(lambda*x+mu)^k*(y(x)-b*x^n-c)^2+b*n*x^(n-1),y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (-2 \left (\cos \left (\lambda x \right ) \cos \left (\mu \right )-\sin \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} x^{n} a b -2 \left (\cos \left (\lambda x \right ) \cos \left (\mu \right )-\sin \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} a c \right ) \left (\cos ^{-k}\left (\lambda x +\mu \right )\right )}{2 a}+\frac {1}{c_{1}-\left (\int \left (\cos \left (\lambda x \right ) \cos \left (\mu \right )-\sin \left (\lambda x \right ) \sin \left (\mu \right )\right )^{k} a d x \right )} \]
✓ Solution by Mathematica
Time used: 2.527 (sec). Leaf size: 92
DSolve[y'[x]==a*Cos[\[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 {\sin ^2(\mu +\lambda x)} \csc (\mu +\lambda x) \cos ^{k+1}(\mu +\lambda x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\cos ^2(x \lambda +\mu )\right )}{(k+1) \lambda }+c_1}+b x^n+c \\ y(x)\to b x^n+c \\ \end{align*}