Internal problem ID [9716]
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]
Solve \begin {gather*} \boxed {y^{\prime }-\left (\left (\cosh ^{2}\left (\lambda x \right )\right ) a -\lambda \right ) y^{2}-a -\lambda +\left (\cosh ^{2}\left (\lambda x \right )\right ) a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (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 \relax (x ) = \frac {\sinh \left (2 \lambda x \right ) \left (-4 \cosh \left (2 \lambda x \right ) \sqrt {\cosh \left (2 \lambda x \right )+1}\, c_{1} a \lambda -4 \sqrt {\cosh \left (2 \lambda x \right )+1}\, c_{1} a \lambda +8 \sqrt {\cosh \left (2 \lambda x \right )+1}\, c_{1} \lambda ^{2}\right ) {\mathrm e}^{\frac {a \cosh \left (2 \lambda x \right )}{2 \lambda }}}{2 \left (\cosh \left (2 \lambda x \right )+1\right )^{2} \sqrt {-1+\cosh \left (2 \lambda x \right )}\, \left (a \left (\cosh ^{2}\left (\lambda x \right )\right )-\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 )}{\left (\cosh \left (2 \lambda x \right )+1\right )^{\frac {3}{2}} \sqrt {-1+\cosh \left (2 \lambda x \right )}}d x \right ) c_{1}+1\right )}+\frac {\sinh \left (2 \lambda x \right ) \left (\left (\left (\cosh ^{2}\left (2 \lambda x \right )\right ) \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 )}{\left (\cosh \left (2 \lambda x \right )+1\right )^{\frac {3}{2}} \sqrt {-1+\cosh \left (2 \lambda x \right )}}d x \right )+a \sqrt {-1+\cosh \left (2 \lambda x \right )}\, \left (\cosh ^{2}\left (2 \lambda x \right )\right )+\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 (\cosh \left (2 \lambda x \right )+1\right )^{2} \sqrt {-1+\cosh \left (2 \lambda x \right )}\, \left (a \left (\cosh ^{2}\left (\lambda x \right )\right )-\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 )}{\left (\cosh \left (2 \lambda x \right )+1\right )^{\frac {3}{2}} \sqrt {-1+\cosh \left (2 \lambda x \right )}}d x \right ) c_{1}+1\right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==(a*Cosh[\[Lambda]*x]^2-\[Lambda])*y[x]^2+a+\[Lambda]-a*Cosh[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
Not solved