61.4.2 problem 23

Internal problem ID [12107]
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
Date solved : Tuesday, January 28, 2025 at 12:24:36 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Riccati]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 64

dsolve(diff(y(x),x)=a*exp(lambda*x)*y(x)^2+b*exp(-lambda*x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.365 (sec). Leaf size: 123

DSolve[D[y[x],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 (4 a b-\lambda \left (\sqrt {\lambda ^2-4 a b}+\lambda \right )\right )}{2 a \sqrt {\lambda ^2-4 a b}} \\ \end{align*}