Internal problem ID [9713]
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]
\[ \boxed {2 y^{\prime }-\left (a -\lambda +a \cosh \left (\lambda x \right )\right ) y^{2}-a -\lambda +a \cosh \left (\lambda x \right )=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 255
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)
\[ y \left (x \right ) = -\frac {2 c_{1} \lambda \sinh \left (\lambda x \right ) {\mathrm e}^{\frac {a \cosh \left (\lambda x \right )}{\lambda }}}{\left (\cosh \left (\lambda x \right )+1\right )^{\frac {3}{2}} \left (\left (\int \frac {\left (a -\lambda +\cosh \left (\lambda x \right ) a \right ) {\mathrm e}^{\frac {a \cosh \left (\lambda x \right )}{\lambda }} \lambda \sinh \left (\lambda x \right )}{\sqrt {\cosh \left (\lambda x \right )-1}\, \left (\cosh \left (\lambda x \right )+1\right )^{\frac {3}{2}}}d x \right ) c_{1} +1\right ) \sqrt {\cosh \left (\lambda x \right )-1}}+\frac {\left (\left (\cosh \left (\lambda x \right ) \sqrt {\cosh \left (\lambda x \right )-1}\, c_{1} +\sqrt {\cosh \left (\lambda x \right )-1}\, c_{1} \right ) \left (\int \frac {\left (a -\lambda +\cosh \left (\lambda x \right ) a \right ) {\mathrm e}^{\frac {a \cosh \left (\lambda x \right )}{\lambda }} \lambda \sinh \left (\lambda x \right )}{\sqrt {\cosh \left (\lambda x \right )-1}\, \left (\cosh \left (\lambda x \right )+1\right )^{\frac {3}{2}}}d x \right )+\cosh \left (\lambda x \right ) \sqrt {\cosh \left (\lambda x \right )-1}+\sqrt {\cosh \left (\lambda x \right )-1}\right ) \sinh \left (\lambda x \right )}{\left (\left (\int \frac {\left (a -\lambda +\cosh \left (\lambda x \right ) a \right ) {\mathrm e}^{\frac {a \cosh \left (\lambda x \right )}{\lambda }} \lambda \sinh \left (\lambda x \right )}{\sqrt {\cosh \left (\lambda x \right )-1}\, \left (\cosh \left (\lambda x \right )+1\right )^{\frac {3}{2}}}d x \right ) c_{1} +1\right ) \sqrt {\cosh \left (\lambda x \right )-1}\, \left (\cosh \left (\lambda x \right )+1\right )^{2}} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[2*y'[x]==(a-\[Lambda]+a*Cosh[\[Lambda]*x])*y[x]^2+a+\[Lambda]-a*Cosh[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
Not solved