61.3.11 problem 11

Internal problem ID [12095]
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
Date solved : Tuesday, January 28, 2025 at 12:22:06 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Riccati]

\begin{align*} y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{2}+b y+c \,{\mathrm e}^{-\lambda x} \end{align*}

✓ Solution by Maple

Time used: 0.008 (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)
 

\[ y = -\frac {\left (-\sqrt {\left (b +\lambda \right )^{2} \left (4 a c -b^{2}-2 b \lambda -\lambda ^{2}\right )}\, \tan \left (\frac {\left (\left (b +\lambda \right ) x +c_{1} \right ) \sqrt {\left (b +\lambda \right )^{2} \left (4 a c -b^{2}-2 b \lambda -\lambda ^{2}\right )}}{2 \left (b +\lambda \right )^{2}}\right )+\left (b +\lambda \right )^{2}\right ) {\mathrm e}^{-\lambda x}}{2 a \left (b +\lambda \right )} \]

✓ Solution by Mathematica

Time used: 0.581 (sec). Leaf size: 188

DSolve[D[y[x],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*}