Internal problem ID [9709]
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: 8.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-\alpha y^{2}-\beta -\gamma \cosh \left (x \right )=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 70
dsolve(diff(y(x),x)=alpha*y(x)^2+beta+gamma*cosh(x),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {i \left (c_{1} \operatorname {MathieuSPrime}\left (-4 \alpha \beta , 2 \gamma \alpha , \frac {i x}{2}\right )+\operatorname {MathieuCPrime}\left (-4 \alpha \beta , 2 \gamma \alpha , \frac {i x}{2}\right )\right )}{2 \alpha \left (c_{1} \operatorname {MathieuS}\left (-4 \alpha \beta , 2 \gamma \alpha , \frac {i x}{2}\right )+\operatorname {MathieuC}\left (-4 \alpha \beta , 2 \gamma \alpha , \frac {i x}{2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.308 (sec). Leaf size: 140
DSolve[y'[x]==\[Alpha]*y[x]^2+\[Beta]+\[Gamma]*Cosh[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {i c_1 \text {MathieuCPrime}\left [-4 \alpha \beta ,2 \alpha \gamma ,\frac {i x}{2}\right ]-i \text {MathieuSPrime}\left [-4 \alpha \beta ,2 \alpha \gamma ,\frac {i x}{2}\right ]}{2 \alpha c_1 \text {MathieuC}\left [-4 \alpha \beta ,2 \alpha \gamma ,\frac {i x}{2}\right ]-2 \alpha \text {MathieuS}\left [-4 \alpha \beta ,2 \alpha \gamma ,\frac {i x}{2}\right ]} \\ y(x)\to -\frac {i \text {MathieuCPrime}\left [-4 \alpha \beta ,2 \alpha \gamma ,\frac {i x}{2}\right ]}{2 \alpha \text {MathieuC}\left [-4 \alpha \beta ,2 \alpha \gamma ,\frac {i x}{2}\right ]} \\ \end{align*}