Internal problem ID [10540]
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: 32.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}-a x \tan \left (b x \right )^{m} y=a \tan \left (b x \right )^{m}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 85
dsolve(diff(y(x),x)=y(x)^2+a*x*tan(b*x)^m*y(x)+a*tan(b*x)^m,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {{\mathrm e}^{\int \frac {a \tan \left (x b \right )^{m} x^{2}-2}{x}d x} x +\int {\mathrm e}^{\int \frac {a \tan \left (x b \right )^{m} x^{2}-2}{x}d x}d x -c_{1}}{x \left (-c_{1} +\int {\mathrm e}^{\int \frac {a \tan \left (x b \right )^{m} x^{2}-2}{x}d x}d x \right )} \]
✓ Solution by Mathematica
Time used: 8.199 (sec). Leaf size: 126
DSolve[y'[x]==y[x]^2+a*x*Tan[b*x]^m*y[x]+a*Tan[b*x]^m,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\exp \left (-\int _1^x-a K[1] \tan ^m(b K[1])dK[1]\right )+x \int _1^x\frac {\exp \left (-\int _1^{K[2]}-a K[1] \tan ^m(b K[1])dK[1]\right )}{K[2]^2}dK[2]+c_1 x}{x^2 \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}-a K[1] \tan ^m(b K[1])dK[1]\right )}{K[2]^2}dK[2]+c_1\right )} y(x)\to -\frac {1}{x} \end{align*}