61.3.6 problem 6

Internal problem ID [12090]
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 : 6
Date solved : Tuesday, January 28, 2025 at 12:21:43 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 73

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

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

✓ Solution by Mathematica

Time used: 0.503 (sec). Leaf size: 115

DSolve[D[y[x],x]==y[x]^2+a*Exp[\[Lambda]*x]*y[x]-a*b*Exp[\[Lambda]*x]-b^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {b \left (-2 \lambda e^{\frac {a e^{\lambda x}}{\lambda }} \left (-\frac {a e^{\lambda x}}{\lambda }\right )^{\frac {2 b}{\lambda }}+2 b \Gamma \left (\frac {2 b}{\lambda },0,-\frac {a e^{x \lambda }}{\lambda }\right )+c_1 \lambda (-1)^{b/\lambda }\right )}{2 b \Gamma \left (\frac {2 b}{\lambda },0,-\frac {a e^{x \lambda }}{\lambda }\right )+c_1 \lambda (-1)^{b/\lambda }} \\ y(x)\to b \\ \end{align*}