Internal problem ID [9714]
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: 9.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-a \cosh \left (\beta x \right ) y-a b \cosh \left (\beta x \right )+b^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 48
dsolve(diff(y(x),x)=y(x)^2+a*cosh(beta*x)*y(x)+a*b*cosh(beta*x)-b^2,y(x), singsol=all)
\[ y \relax (x ) = -b -\frac {{\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }-2 b x}}{\int {\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }-2 b x}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 4.455 (sec). Leaf size: 189
DSolve[y'[x]==y[x]^2+a*Cosh[\[Beta]*x]*y[x]+a*b*Cosh[\[Beta]*x]-b^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -b-\frac {\beta \left (e^{\beta x}\right )^{-\frac {2 b}{\beta }} \left (\sinh \left (\frac {a \sinh (\beta x)}{\beta }\right )+\cosh \left (\frac {a \sinh (\beta x)}{\beta }\right )\right )}{\int _1^{e^{x \beta }}e^{\frac {a \left (K[1]^2-1\right )}{2 \beta K[1]}} K[1]^{-\frac {2 b}{\beta }-1}dK[1]+c_1} \\ y(x)\to -b \\ y(x)\to -\frac {\beta \left (e^{\beta x}\right )^{-\frac {2 b}{\beta }} \left (\sinh \left (\frac {a \sinh (\beta x)}{\beta }\right )+\cosh \left (\frac {a \sinh (\beta x)}{\beta }\right )\right )}{\int _1^{e^{x \beta }}e^{\frac {a \left (K[1]^2-1\right )}{2 \beta K[1]}} K[1]^{-\frac {2 b}{\beta }-1}dK[1]}-b \\ \end{align*}