Internal problem ID [9715]
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: 10.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-a x \left (\cosh ^{m}\left (b x \right )\right ) y-a \left (\cosh ^{m}\left (b x \right )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 85
dsolve(diff(y(x),x)=y(x)^2+a*x*cosh(b*x)^m*y(x)+a*cosh(b*x)^m,y(x), singsol=all)
\[ y \relax (x ) = -\frac {{\mathrm e}^{\int \frac {a \left (\cosh ^{m}\left (b x \right )\right ) x^{2}-2}{x}d x} x +\int {\mathrm e}^{\int \frac {a \left (\cosh ^{m}\left (b x \right )\right ) x^{2}-2}{x}d x}d x -c_{1}}{\left (-c_{1}+\int {\mathrm e}^{\int \frac {a \left (\cosh ^{m}\left (b x \right )\right ) x^{2}-2}{x}d x}d x \right ) x} \]
✓ Solution by Mathematica
Time used: 53.771 (sec). Leaf size: 247
DSolve[y'[x]==y[x]^2+a*x*Cosh[b*x]^m*y[x]+a*Cosh[b*x]^m,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x+\frac {\exp \left (\frac {a \cosh ^{m+1}(b x) \left (\frac {\cosh (b x) \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;\cosh ^2(b x)\right )}{m+2}-b x \sinh (b x) \, _2F_1\left (1,\frac {m+2}{2};\frac {m+3}{2};\cosh ^2(b x)\right )\right )}{b^2 (m+1)}\right )}{\int _1^x\frac {\exp \left (\frac {a \cosh ^{m+1}(b K[1]) \left (\frac {\cosh (b K[1]) \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;\cosh ^2(b K[1])\right )}{m+2}-b \, _2F_1\left (1,\frac {m+2}{2};\frac {m+3}{2};\cosh ^2(b K[1])\right ) K[1] \sinh (b K[1])\right )}{b^2 (m+1)}\right )}{K[1]^2}dK[1]+c_1}}{x^2} \\ y(x)\to -\frac {1}{x} \\ \end{align*}