Internal problem ID [10429]
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]
\[ \boxed {y^{\prime }-a \,{\mathrm e}^{\lambda x} y^{2}-b y=c \,{\mathrm e}^{-\lambda x}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 165
dsolve(diff(y(x),x)=a*exp(lambda*x)*y(x)^2+b*y(x)+c*exp(-lambda*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {{\mathrm e}^{-\lambda x} \left (\sqrt {4 b^{2} a c +8 a b c \lambda +4 \lambda ^{2} a c -b^{4}-4 b^{3} \lambda -6 b^{2} \lambda ^{2}-4 b \,\lambda ^{3}-\lambda ^{4}}\, \tan \left (\frac {\sqrt {4 b^{2} a c +8 a b c \lambda +4 \lambda ^{2} a c -b^{4}-4 b^{3} \lambda -6 b^{2} \lambda ^{2}-4 b \,\lambda ^{3}-\lambda ^{4}}\, \left (x b +\lambda x +c_{1} \right )}{2 b^{2}+4 b \lambda +2 \lambda ^{2}}\right )-b^{2}-2 b \lambda -\lambda ^{2}\right )}{2 a \left (b +\lambda \right )} \]
✓ 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*}