Internal problem ID [9708]
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]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-a x \left (\sinh ^{m}\left (x b \right )\right ) y-a \left (\sinh ^{m}\left (x b \right )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (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)
\[ y \relax (x ) = -\frac {{\mathrm e}^{\int \frac {a \left (\sinh ^{m}\left (b x \right )\right ) x^{2}-2}{x}d x} x +\int {\mathrm e}^{\int \frac {a \left (\sinh ^{m}\left (b x \right )\right ) x^{2}-2}{x}d x}d x -c_{1}}{\left (-c_{1}+\int {\mathrm e}^{\int \frac {a \left (\sinh ^{m}\left (b x \right )\right ) x^{2}-2}{x}d x}d x \right ) x} \]
✓ Solution by Mathematica
Time used: 4.007 (sec). Leaf size: 232
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 {x+\frac {\exp \left (-\frac {a \left (1-e^{2 b x}\right )^{-m} \sinh ^m(b x) \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 \, _2F_1\left (-m,-\frac {m}{2};1-\frac {m}{2};e^{2 b x}\right )\right )}{b^2 m^2}\right )}{\int _1^x\frac {\exp \left (-\frac {a \left (1-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 \, _2F_1\left (-m,-\frac {m}{2};1-\frac {m}{2};e^{2 b K[1]}\right ) K[1]\right ) \sinh ^m(b K[1])}{b^2 m^2}\right )}{K[1]^2}dK[1]+c_1}}{x^2} \\ y(x)\to -\frac {1}{x} \\ \end{align*}