Internal problem ID [10459]
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: 11.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-\left (a \cosh \left (\lambda x \right )^{2}-\lambda \right ) y^{2}=-a \cosh \left (\lambda x \right )^{2}+a +\lambda } \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 104
dsolve(diff(y(x),x)=(a*cosh(lambda*x)^2-lambda)*y(x)^2+a+lambda-a*cosh(lambda*x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {2 \tanh \left (x \lambda \right ) \lambda \left (\int -{\mathrm e}^{\frac {a \cosh \left (2 x \lambda \right )}{2 \lambda }} \left (a -\operatorname {sech}\left (x \lambda \right )^{2} \lambda \right )d x \right ) c_{1} +2 \operatorname {sech}\left (x \lambda \right )^{2} {\mathrm e}^{\frac {a \cosh \left (2 x \lambda \right )}{2 \lambda }} c_{1} \lambda -\tanh \left (x \lambda \right )}{2 \lambda \left (\int -{\mathrm e}^{\frac {a \cosh \left (2 x \lambda \right )}{2 \lambda }} \left (a -\operatorname {sech}\left (x \lambda \right )^{2} \lambda \right )d x \right ) c_{1} -1} \]
✓ Solution by Mathematica
Time used: 49.81 (sec). Leaf size: 211
DSolve[y'[x]==(a*Cosh[\[Lambda]*x]^2-\[Lambda])*y[x]^2+a+\[Lambda]-a*Cosh[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\text {sech}^2(\lambda x) \left (c_1 \sinh (2 \lambda x) \int _1^xe^{\frac {a \cosh ^2(\lambda K[1])}{\lambda }} \left (\lambda -a \cosh ^2(\lambda K[1])\right ) \text {sech}^2(\lambda K[1])dK[1]+2 c_1 e^{\frac {a \cosh ^2(\lambda x)}{\lambda }}+\sinh (2 \lambda x)\right )}{2+2 c_1 \int _1^xe^{\frac {a \cosh ^2(\lambda K[1])}{\lambda }} \left (\lambda -a \cosh ^2(\lambda K[1])\right ) \text {sech}^2(\lambda K[1])dK[1]} \\ y(x)\to \frac {1}{2} \text {sech}^2(\lambda x) \left (\frac {2 e^{\frac {a \cosh ^2(\lambda x)}{\lambda }}}{\int _1^xe^{\frac {a \cosh ^2(\lambda K[1])}{\lambda }} \left (\lambda -a \cosh ^2(\lambda K[1])\right ) \text {sech}^2(\lambda K[1])dK[1]}+\sinh (2 \lambda x)\right ) \\ \end{align*}