61.5.8 problem 8

Internal problem ID [12132]
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
Date solved : Tuesday, January 28, 2025 at 12:29:11 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 65

dsolve(diff(y(x),x)=alpha*y(x)^2+beta+gamma*cosh(x),y(x), singsol=all)
 

\[ y = -\frac {i \left (c_{1} \operatorname {MathieuSPrime}\left (-4 \alpha \beta , 2 \alpha \gamma , \frac {i x}{2}\right )+\operatorname {MathieuCPrime}\left (-4 \alpha \beta , 2 \alpha \gamma , \frac {i x}{2}\right )\right )}{2 \alpha \left (c_{1} \operatorname {MathieuS}\left (-4 \alpha \beta , 2 \alpha \gamma , \frac {i x}{2}\right )+\operatorname {MathieuC}\left (-4 \alpha \beta , 2 \alpha \gamma , \frac {i x}{2}\right )\right )} \]

✓ Solution by Mathematica

Time used: 0.257 (sec). Leaf size: 140

DSolve[D[y[x],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*}