Internal problem ID [9684]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.3-2. Equations with power and
exponential functions
Problem number: 23.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _Riccati]
\[ \boxed {y^{\prime }-a \,{\mathrm e}^{\lambda x} y^{2}-{\mathrm e}^{-\lambda x} b=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 75
dsolve(diff(y(x),x)=a*exp(lambda*x)*y(x)^2+b*exp(-lambda*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-{\mathrm e}^{\lambda x} {\mathrm e}^{-\lambda x} \lambda ^{2}+\tan \left (\frac {\sqrt {4 a b \,\lambda ^{2}-\lambda ^{4}}\, \left (\lambda x +c_{1} \right )}{2 \lambda ^{2}}\right ) \sqrt {4 a b \,\lambda ^{2}-\lambda ^{4}}\right ) {\mathrm e}^{-\lambda x}}{2 a \lambda } \]
✓ Solution by Mathematica
Time used: 0.396 (sec). Leaf size: 103
DSolve[y'[x]==a*Exp[\[Lambda]*x]*y[x]^2+b*Exp[-\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{\lambda (-x)} \left (-\sqrt {\lambda ^2-4 a b}+\frac {2}{\frac {1}{\sqrt {\lambda ^2-4 a b}}+c_1 e^{x \sqrt {\lambda ^2-4 a b}}}-\lambda \right )}{2 a} \\ y(x)\to -\frac {e^{\lambda (-x)} \left (\sqrt {\lambda ^2-4 a b}+\lambda \right )}{2 a} \\ \end{align*}