Internal problem ID [9790]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-3. Equations with tangent.
Problem number: 35.
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 (\tan ^{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: 134
dsolve(diff(y(x),x)=a*tan(lambda*x+mu)^k*(y(x)-b*x^n-c)^2+b*n*x^(n-1),y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (-2 x^{n} \left (\frac {-\tan \left (\mu \right )-\tan \left (\lambda x \right )}{\tan \left (\mu \right ) \tan \left (\lambda x \right )-1}\right )^{k} a b -2 \left (\frac {-\tan \left (\mu \right )-\tan \left (\lambda x \right )}{\tan \left (\mu \right ) \tan \left (\lambda x \right )-1}\right )^{k} a c \right ) \left (-\frac {\tan \left (\mu \right )+\tan \left (\lambda x \right )}{\tan \left (\mu \right ) \tan \left (\lambda x \right )-1}\right )^{-k}}{2 a}+\frac {1}{c_{1}-\left (\int \left (\frac {-\tan \left (\mu \right )-\tan \left (\lambda x \right )}{\tan \left (\mu \right ) \tan \left (\lambda x \right )-1}\right )^{k} a d x \right )} \]
✓ Solution by Mathematica
Time used: 2.647 (sec). Leaf size: 75
DSolve[y'[x]==a*Tan[\[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 \tan ^{k+1}(\mu +\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};-\tan ^2(\mu +x \lambda )\right )}{(k+1) \lambda }+c_1}+b x^n+c \\ y(x)\to b x^n+c \\ \end{align*}