61.10.10 problem 23

Internal problem ID [12197]
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
Date solved : Tuesday, January 28, 2025 at 01:03:22 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} y^{\prime }&=a \cos \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: 38

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)
 

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

✓ Solution by Mathematica

Time used: 1.089 (sec). Leaf size: 92

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