Internal problem ID [10461]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.4-1. Equations with hyperbolic
sine and cosine
Problem number: 3.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}-a x \sinh \left (b x \right )^{m} y=a \sinh \left (b x \right )^{m}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 85
dsolve(diff(y(x),x)=y(x)^2+a*x*sinh(b*x)^m*y(x)+a*sinh(b*x)^m,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {{\mathrm e}^{\int \frac {a \sinh \left (x b \right )^{m} x^{2}-2}{x}d x} x +\int {\mathrm e}^{\int \frac {a \sinh \left (x b \right )^{m} x^{2}-2}{x}d x}d x -c_{1}}{\left (-c_{1} +\int {\mathrm e}^{\int \frac {a \sinh \left (x b \right )^{m} x^{2}-2}{x}d x}d x \right ) x} \]
✓ Solution by Mathematica
Time used: 7.437 (sec). Leaf size: 379
DSolve[y'[x]==y[x]^2+a*x*Sinh[b*x]^m*y[x]+a*Sinh[b*x]^m,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\int _1^x\frac {\exp \left (-\frac {a \left (-e^{-b K[1]}+e^{b K[1]}\right )^m \left (2-2 e^{2 b K[1]}\right )^{-m} \left (\, _3F_2\left (-m,-\frac {m}{2},-\frac {m}{2};1-\frac {m}{2},1-\frac {m}{2};e^{2 b K[1]}\right )+b m \operatorname {Hypergeometric2F1}\left (-m,-\frac {m}{2},1-\frac {m}{2},e^{2 b K[1]}\right ) K[1]\right )}{b^2 m^2}\right )}{K[1]^2}dK[1]+\frac {\exp \left (-\frac {a \left (e^{b x}-e^{-b x}\right )^m \left (2-2 e^{2 b x}\right )^{-m} \left (\, _3F_2\left (-m,-\frac {m}{2},-\frac {m}{2};1-\frac {m}{2},1-\frac {m}{2};e^{2 b x}\right )+b m x \operatorname {Hypergeometric2F1}\left (-m,-\frac {m}{2},1-\frac {m}{2},e^{2 b x}\right )\right )}{b^2 m^2}\right )}{x}+c_1}{x \left (\int _1^x\frac {\exp \left (-\frac {a \left (-e^{-b K[1]}+e^{b K[1]}\right )^m \left (2-2 e^{2 b K[1]}\right )^{-m} \left (\, _3F_2\left (-m,-\frac {m}{2},-\frac {m}{2};1-\frac {m}{2},1-\frac {m}{2};e^{2 b K[1]}\right )+b m \operatorname {Hypergeometric2F1}\left (-m,-\frac {m}{2},1-\frac {m}{2},e^{2 b K[1]}\right ) K[1]\right )}{b^2 m^2}\right )}{K[1]^2}dK[1]+c_1\right )} y(x)\to -\frac {1}{x} \end{align*}