Internal problem ID [10469]
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: 464
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 {\sinh \left (2 \lambda x \right ) \left (-4 \cosh \left (2 \lambda x \right ) \sqrt {1+\cosh \left (2 \lambda x \right )}\, c_{1} a \lambda -4 \sqrt {1+\cosh \left (2 \lambda x \right )}\, c_{1} a \lambda +8 \sqrt {1+\cosh \left (2 \lambda x \right )}\, c_{1} \lambda ^{2}\right ) {\mathrm e}^{\frac {a \cosh \left (2 \lambda x \right )}{2 \lambda }}}{2 \left (1+\cosh \left (2 \lambda x \right )\right )^{2} \sqrt {-1+\cosh \left (2 \lambda x \right )}\, \left (a \cosh \left (\lambda x \right )^{2}-\lambda \right ) \left (\left (\int \frac {2 \left (a \cosh \left (2 \lambda x \right )+a -2 \lambda \right ) {\mathrm e}^{\frac {a \cosh \left (2 \lambda x \right )}{2 \lambda }} \lambda \sinh \left (2 \lambda x \right )}{\sqrt {-1+\cosh \left (2 \lambda x \right )}\, \left (1+\cosh \left (2 \lambda x \right )\right )^{\frac {3}{2}}}d x \right ) c_{1} +1\right )}+\frac {\sinh \left (2 \lambda x \right ) \left (\left (\cosh \left (2 \lambda x \right )^{2} \sqrt {-1+\cosh \left (2 \lambda x \right )}\, c_{1} a +\left (2 \sqrt {-1+\cosh \left (2 \lambda x \right )}\, c_{1} a -2 \sqrt {-1+\cosh \left (2 \lambda x \right )}\, c_{1} \lambda \right ) \cosh \left (2 \lambda x \right )+\sqrt {-1+\cosh \left (2 \lambda x \right )}\, c_{1} a -2 \sqrt {-1+\cosh \left (2 \lambda x \right )}\, c_{1} \lambda \right ) \left (\int \frac {2 \left (a \cosh \left (2 \lambda x \right )+a -2 \lambda \right ) {\mathrm e}^{\frac {a \cosh \left (2 \lambda x \right )}{2 \lambda }} \lambda \sinh \left (2 \lambda x \right )}{\sqrt {-1+\cosh \left (2 \lambda x \right )}\, \left (1+\cosh \left (2 \lambda x \right )\right )^{\frac {3}{2}}}d x \right )+a \sqrt {-1+\cosh \left (2 \lambda x \right )}\, \cosh \left (2 \lambda x \right )^{2}+\left (2 a \sqrt {-1+\cosh \left (2 \lambda x \right )}-2 \lambda \sqrt {-1+\cosh \left (2 \lambda x \right )}\right ) \cosh \left (2 \lambda x \right )+a \sqrt {-1+\cosh \left (2 \lambda x \right )}-2 \lambda \sqrt {-1+\cosh \left (2 \lambda x \right )}\right )}{2 \left (1+\cosh \left (2 \lambda x \right )\right )^{2} \sqrt {-1+\cosh \left (2 \lambda x \right )}\, \left (a \cosh \left (\lambda x \right )^{2}-\lambda \right ) \left (\left (\int \frac {2 \left (a \cosh \left (2 \lambda x \right )+a -2 \lambda \right ) {\mathrm e}^{\frac {a \cosh \left (2 \lambda x \right )}{2 \lambda }} \lambda \sinh \left (2 \lambda x \right )}{\sqrt {-1+\cosh \left (2 \lambda x \right )}\, \left (1+\cosh \left (2 \lambda x \right )\right )^{\frac {3}{2}}}d x \right ) c_{1} +1\right )} \]
✓ 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*}