Internal problem ID [9677]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.3. Equations Containing Exponential
Functions
Problem number: 12.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-a \,{\mathrm e}^{\mu x} y^{2}-\lambda y+a \,b^{2} {\mathrm e}^{\left (\mu +2 \lambda \right ) x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 86
dsolve(diff(y(x),x)=a*exp(mu*x)*y(x)^2+lambda*y(x)-a*b^2*exp((mu+2*lambda)*x),y(x), singsol=all)
\[ y \relax (x ) = -\frac {b \left (c_{1} \sinh \left (\frac {b a \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )+\cosh \left (\frac {b a \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )\right ) {\mathrm e}^{x \left (\lambda +\mu \right )-\mu x}}{c_{1} \cosh \left (\frac {b a \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )+\sinh \left (\frac {b a \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )} \]
✓ Solution by Mathematica
Time used: 0.698 (sec). Leaf size: 282
DSolve[y'[x]==a*Exp[\[Mu]*x]*y[x]^2+\[Lambda]*y[x]-a*b^2*Exp[(\[Mu]+2*\[Lambda])*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \fbox {$-\frac {\tan \left (\frac {a b^2 e^{x (2 \lambda +\mu )} \sqrt {-\frac {e^{-2 x \lambda }}{b^2}}}{\lambda +\mu }-c_1\right )}{\sqrt {-\frac {e^{-2 x \lambda }}{b^2}}}\text { if }\text {condition}$} \\ \end{align*}