61.5.9 problem 9

Internal problem ID [12133]
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
Date solved : Tuesday, January 28, 2025 at 12:29:14 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}+a \cosh \left (\beta x \right ) y+a b \cosh \left (\beta x \right )-b^{2} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 73

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)
 

\[ y = \frac {b \left (\int {\mathrm e}^{\frac {-2 b \beta x +a \sinh \left (\beta x \right )}{\beta }}d x \right )-c_{1} b +{\mathrm e}^{\frac {-2 b \beta x +a \sinh \left (\beta x \right )}{\beta }}}{-\int {\mathrm e}^{\frac {-2 b \beta x +a \sinh \left (\beta x \right )}{\beta }}d x +c_{1}} \]

✓ Solution by Mathematica

Time used: 4.242 (sec). Leaf size: 298

DSolve[D[y[x],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 -\frac {\beta \exp \left (\beta x-2 \int _1^{e^{x \beta }}\frac {2 (2 b+\beta ) K[1]-a \left (K[1]^2+1\right )}{4 \beta K[1]^2}dK[1]\right )+b \int _1^{e^{x \beta }}\exp \left (-2 \int _1^{K[3]}\frac {2 (2 b+\beta ) K[1]-a \left (K[1]^2+1\right )}{4 \beta K[1]^2}dK[1]\right )dK[3]+b c_1}{\int _1^{e^{x \beta }}\exp \left (-2 \int _1^{K[3]}\frac {2 (2 b+\beta ) K[1]-a \left (K[1]^2+1\right )}{4 \beta K[1]^2}dK[1]\right )dK[3]+c_1} \\ y(x)\to -b \\ y(x)\to -\frac {\beta \exp \left (\beta x-2 \int _1^{e^{x \beta }}\frac {2 (2 b+\beta ) K[1]-a \left (K[1]^2+1\right )}{4 \beta K[1]^2}dK[1]\right )}{\int _1^{e^{x \beta }}\exp \left (-2 \int _1^{K[3]}\frac {2 (2 b+\beta ) K[1]-a \left (K[1]^2+1\right )}{4 \beta K[1]^2}dK[1]\right )dK[3]}-b \\ \end{align*}