Internal problem ID [10543]
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]
\[ \boxed {y^{\prime }-a \tan \left (x \lambda +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}=b n \,x^{-1+n}} \]
✓ Solution by Maple
Time used: 0.016 (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)
\[ y \left (x \right ) = -\frac {\left (-2 a \,x^{n} \left (\frac {-\tan \left (\mu \right )-\tan \left (\lambda x \right )}{\tan \left (\mu \right ) \tan \left (\lambda x \right )-1}\right )^{k} b -2 a c \left (\frac {-\tan \left (\mu \right )-\tan \left (\lambda x \right )}{\tan \left (\mu \right ) \tan \left (\lambda x \right )-1}\right )^{k}\right ) \left (-\frac {\tan \left (\mu \right )+\tan \left (\lambda x \right )}{\tan \left (\mu \right ) \tan \left (\lambda x \right )-1}\right )^{-k}}{2 a}+\frac {1}{c_{1} -\left (\int a \left (\frac {-\tan \left (\mu \right )-\tan \left (\lambda x \right )}{\tan \left (\mu \right ) \tan \left (\lambda x \right )-1}\right )^{k}d x \right )} \]
✓ Solution by Mathematica
Time used: 6.024 (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) \operatorname {Hypergeometric2F1}\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*}