Internal problem ID [10460]
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: 12.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {2 y^{\prime }-\left (a -\lambda +a \cosh \left (\lambda x \right )\right ) y^{2}=a +\lambda -a \cosh \left (\lambda x \right )} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 101
dsolve(2*diff(y(x),x)=(a-lambda+a*cosh(lambda*x))*y(x)^2+a+lambda-a*cosh(lambda*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\tanh \left (\frac {x \lambda }{2}\right ) \lambda \left (\int {\mathrm e}^{\frac {a \cosh \left (x \lambda \right )}{\lambda }} \left (-2 a +\operatorname {sech}\left (\frac {x \lambda }{2}\right )^{2} \lambda \right )d x \right ) c_{1} +2 \operatorname {sech}\left (\frac {x \lambda }{2}\right )^{2} {\mathrm e}^{\frac {a \cosh \left (x \lambda \right )}{\lambda }} c_{1} \lambda -2 \tanh \left (\frac {x \lambda }{2}\right )}{\lambda \left (\int {\mathrm e}^{\frac {a \cosh \left (x \lambda \right )}{\lambda }} \left (-2 a +\operatorname {sech}\left (\frac {x \lambda }{2}\right )^{2} \lambda \right )d x \right ) c_{1} -2} \]
✓ Solution by Mathematica
Time used: 59.899 (sec). Leaf size: 338
DSolve[2*y'[x]==(a-\[Lambda]+a*Cosh[\[Lambda]*x])*y[x]^2+a+\[Lambda]-a*Cosh[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\text {sech}^2\left (\frac {\lambda x}{2}\right ) \left (c_1 \sinh (\lambda x) \int _1^x-e^{\frac {2 a \cosh ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} (\cosh (\lambda K[1]) a+a-\lambda ) \text {sech}^2\left (\frac {1}{2} \lambda K[1]\right )dK[1]+4 c_1 e^{\frac {2 a \cosh ^2\left (\frac {\lambda x}{2}\right )}{\lambda }}+\sinh (\lambda x)\right )}{2+2 c_1 \int _1^x-e^{\frac {2 a \cosh ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} (\cosh (\lambda K[1]) a+a-\lambda ) \text {sech}^2\left (\frac {1}{2} \lambda K[1]\right )dK[1]} \\ y(x)\to \frac {1}{2} \text {sech}^2\left (\frac {\lambda x}{2}\right ) \left (\frac {4 e^{\frac {2 a \cosh ^2\left (\frac {\lambda x}{2}\right )}{\lambda }}}{\int _1^x-e^{\frac {2 a \cosh ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} (\cosh (\lambda K[1]) a+a-\lambda ) \text {sech}^2\left (\frac {1}{2} \lambda K[1]\right )dK[1]}+\sinh (\lambda x)\right ) \\ y(x)\to \frac {1}{2} \text {sech}^2\left (\frac {\lambda x}{2}\right ) \left (\frac {4 e^{\frac {2 a \cosh ^2\left (\frac {\lambda x}{2}\right )}{\lambda }}}{\int _1^x-e^{\frac {2 a \cosh ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} (\cosh (\lambda K[1]) a+a-\lambda ) \text {sech}^2\left (\frac {1}{2} \lambda K[1]\right )dK[1]}+\sinh (\lambda x)\right ) \\ \end{align*}