Internal problem ID [10419]
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: 11.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _Riccati]
\[ \boxed {y^{\prime }-a \,{\mathrm e}^{\lambda x} y^{2}-y b=c \,{\mathrm e}^{-\lambda x}} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 96
dsolve(diff(y(x),x)=a*exp(lambda*x)*y(x)^2+b*y(x)+c*exp(-lambda*x),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (-\sqrt {\left (b +\lambda \right )^{2} \left (4 a c -b^{2}-2 \lambda b -\lambda ^{2}\right )}\, \tan \left (\frac {\left (\left (b +\lambda \right ) x +c_{1} \right ) \sqrt {\left (b +\lambda \right )^{2} \left (4 a c -b^{2}-2 \lambda b -\lambda ^{2}\right )}}{2 \left (b +\lambda \right )^{2}}\right )+\left (b +\lambda \right )^{2}\right ) {\mathrm e}^{-x \lambda }}{2 a \left (b +\lambda \right )} \]
✓ Solution by Mathematica
Time used: 0.927 (sec). Leaf size: 188
DSolve[y'[x]==a*Exp[\[Lambda]*x]*y[x]^2+b*y[x]+c*Exp[-\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{\lambda (-x)} \left (-\sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}+\frac {2}{\frac {1}{\sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}}+c_1 e^{x \sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}}}-b-\lambda \right )}{2 a} \\ y(x)\to -\frac {e^{\lambda (-x)} \left (b \left (\sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}+2 \lambda \right )+\lambda \left (\sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}+\lambda \right )-4 a c+b^2\right )}{2 a \sqrt {-4 a c+b^2+2 b \lambda +\lambda ^2}} \\ \end{align*}