Internal problem ID [9676]
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]
Solve \begin {gather*} \boxed {y^{\prime }-a \,{\mathrm e}^{\lambda x} y^{2}-b y-c \,{\mathrm e}^{-\lambda x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (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)
\[ y \relax (x ) = \frac {{\mathrm e}^{-\lambda x} \left (\sqrt {4 a \,b^{2} c +8 a b c \lambda +4 \lambda ^{2} c a -b^{4}-4 b^{3} \lambda -6 b^{2} \lambda ^{2}-4 b \,\lambda ^{3}-\lambda ^{4}}\, \tan \left (\frac {\sqrt {4 a \,b^{2} c +8 a b c \lambda +4 \lambda ^{2} c a -b^{4}-4 b^{3} \lambda -6 b^{2} \lambda ^{2}-4 b \,\lambda ^{3}-\lambda ^{4}}\, \left (b x +\lambda x +c_{1}\right )}{2 b^{2}+4 b \lambda +2 \lambda ^{2}}\right )-b^{2}-2 b \lambda -\lambda ^{2}\right )}{2 a \left (b +\lambda \right )} \]
✓ Solution by Mathematica
Time used: 0.525 (sec). Leaf size: 115
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 {(b+\lambda )^2-4 a c}+\frac {2}{\frac {1}{\sqrt {(b+\lambda )^2-4 a c}}+c_1 e^{x \sqrt {(b+\lambda )^2-4 a c}}}-b-\lambda \right )}{2 a} \\ y(x)\to -\frac {e^{\lambda (-x)} \left (\sqrt {(b+\lambda )^2-4 a c}+b+\lambda \right )}{2 a} \\ \end{align*}