Internal problem ID [9667]
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: 2.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-a \lambda \,{\mathrm e}^{\lambda x}+a^{2} {\mathrm e}^{2 \lambda x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 63
dsolve(diff(y(x),x)=y(x)^2+a*lambda*exp(lambda*x)-a^2*exp(2*lambda*x),y(x), singsol=all)
\[ y \relax (x ) = \frac {{\mathrm e}^{\lambda x} \expIntegral \left (1, -\frac {2 a \,{\mathrm e}^{\lambda x}}{\lambda }\right ) c_{1} a +{\mathrm e}^{\frac {2 a \,{\mathrm e}^{\lambda x}}{\lambda }} c_{1} \lambda +{\mathrm e}^{\lambda x} a}{\expIntegral \left (1, -\frac {2 a \,{\mathrm e}^{\lambda x}}{\lambda }\right ) c_{1}+1} \]
✓ Solution by Mathematica
Time used: 1.722 (sec). Leaf size: 57
DSolve[y'[x]==y[x]^2+a*\[Lambda]*Exp[\[Lambda]*x]-a^2*Exp[2*\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to a e^{\lambda x}-\frac {\lambda e^{\frac {2 a e^{\lambda x}}{\lambda }}}{\text {Ei}\left (\frac {2 a e^{x \lambda }}{\lambda }\right )+c_1} \\ y(x)\to a e^{\lambda x} \\ \end{align*}