Internal problem ID [9784]
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: 33.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }+\left (1+k \right ) x^{k} y^{2}-a \,x^{1+k} \tan \left (x \right )^{m} y+a \tan \left (x \right )^{m}=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 172
dsolve(diff(y(x),x)=-(k+1)*x^k*y(x)^2+a*x^(k+1)*tan(x)^m*y(x)-a*tan(x)^m,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left ({\mathrm e}^{\int \frac {x^{k} \tan \left (x \right )^{m} a \,x^{2}-2 k -2}{x}d x} x \,x^{k}+\left (\int x^{k} {\mathrm e}^{\int \frac {a \,x^{k +2} \tan \left (x \right )^{m}-2 k -2}{x}d x}d x \right ) k +\int x^{k} {\mathrm e}^{\int \frac {a \,x^{k +2} \tan \left (x \right )^{m}-2 k -2}{x}d x}d x -c_{1} \right ) x^{-k}}{x \left (\left (\int x^{k} {\mathrm e}^{\int \frac {a \,x^{k +2} \tan \left (x \right )^{m}-2 k -2}{x}d x}d x \right ) k +\int x^{k} {\mathrm e}^{\int \frac {a \,x^{k +2} \tan \left (x \right )^{m}-2 k -2}{x}d x}d x -c_{1} \right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==-(k+1)*x^k*y[x]^2+a*x^(k+1)*Tan[x]^m*y[x]-a*Tan[x]^m,y[x],x,IncludeSingularSolutions -> True]
Not solved