Internal problem ID [9671]
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: 6.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-a \,{\mathrm e}^{\lambda x} y+{\mathrm e}^{\lambda x} a b +b^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 46
dsolve(diff(y(x),x)=y(x)^2+a*exp(lambda*x)*y(x)-a*b*exp(lambda*x)-b^2,y(x), singsol=all)
\[ y \relax (x ) = b -\frac {{\mathrm e}^{\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }+2 b x}}{\int {\mathrm e}^{\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }+2 b x}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 0.911 (sec). Leaf size: 82
DSolve[y'[x]==y[x]^2+a*Exp[\[Lambda]*x]*y[x]-a*b*Exp[\[Lambda]*x]-b^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to b-\frac {2 b \lambda e^{\frac {a e^{\lambda x}}{\lambda }} \left (-\frac {a e^{\lambda x}}{\lambda }\right )^{\frac {2 b}{\lambda }}}{2 b \Gamma \left (\frac {2 b}{\lambda },0,-\frac {a e^{x \lambda }}{\lambda }\right )+c_1 \lambda (-1)^{b/\lambda }} \\ y(x)\to b \\ \end{align*}