Internal problem ID [9680]
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: 19.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]
\[ \boxed {y^{\prime }-{\mathrm e}^{\mu x} \left (y-b \,{\mathrm e}^{\lambda x}\right )^{2}-b \lambda \,{\mathrm e}^{\lambda x}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 74
dsolve(diff(y(x),x)=exp(mu*x)*(y(x)-b*exp(lambda*x))^2+b*lambda*exp(lambda*x),y(x), singsol=all)
\[ y \left (x \right ) = \left ({\mathrm e}^{x \left (\lambda +\mu \right )} b -\mu +\frac {{\mathrm e}^{-\mu x -\frac {2 b \,{\mathrm e}^{\lambda x +\mu x}}{\lambda +\mu }+\frac {2 \,{\mathrm e}^{x \left (\lambda +\mu \right )} b}{\lambda +\mu }}}{c_{1} +\frac {{\mathrm e}^{-\mu x}}{\mu }}\right ) {\mathrm e}^{-\mu x} \]
✓ Solution by Mathematica
Time used: 0.975 (sec). Leaf size: 40
DSolve[y'[x]==Exp[\[Mu]*x]*(y[x]-b*Exp[\[Lambda]*x])^2+b*\[Lambda]*Exp[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to b e^{\lambda x}+\frac {\mu }{-e^{\mu x}+c_1 \mu } \\ y(x)\to b e^{\lambda x} \\ \end{align*}