Internal problem ID [9670]
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: 5.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-b y-a \left (\lambda -b \right ) {\mathrm e}^{\lambda x}+a^{2} {\mathrm e}^{2 \lambda x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 114
dsolve(diff(y(x),x)=y(x)^2+b*y(x)+a*(lambda-b)*exp(lambda*x)-a^2*exp(2*lambda*x),y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (\left (\int {\mathrm e}^{b x +\frac {2 a \,{\mathrm e}^{\lambda x}}{\lambda }}d x \right ) a +c_{1} a \right ) {\mathrm e}^{\frac {2 a \,{\mathrm e}^{\lambda x}}{\lambda }} {\mathrm e}^{\lambda x -\frac {2 a \,{\mathrm e}^{\lambda x}}{\lambda }}}{\int {\mathrm e}^{b x +\frac {2 a \,{\mathrm e}^{\lambda x}}{\lambda }}d x +c_{1}}-\frac {{\mathrm e}^{b x} {\mathrm e}^{\frac {2 a \,{\mathrm e}^{\lambda x}}{\lambda }}}{\int {\mathrm e}^{b x +\frac {2 a \,{\mathrm e}^{\lambda x}}{\lambda }}d x +c_{1}} \]
✓ Solution by Mathematica
Time used: 1.506 (sec). Leaf size: 174
DSolve[y'[x]==y[x]^2+b*y[x]+a*(\[Lambda]-b)*Exp[\[Lambda]*x]-a^2*Exp[2*\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2^{b/\lambda } \left (\frac {a e^{\lambda x}}{\lambda }\right )^{b/\lambda } \left (\left (a e^{\lambda x}-b\right ) L_{-\frac {b}{\lambda }}^{\frac {b}{\lambda }}\left (\frac {2 a e^{x \lambda }}{\lambda }\right )+2 a e^{\lambda x} L_{-\frac {b+\lambda }{\lambda }}^{\frac {b+\lambda }{\lambda }}\left (\frac {2 a e^{x \lambda }}{\lambda }\right )\right )+a c_1 e^{\lambda x}}{2^{b/\lambda } \left (\frac {a e^{\lambda x}}{\lambda }\right )^{b/\lambda } L_{-\frac {b}{\lambda }}^{\frac {b}{\lambda }}\left (\frac {2 a e^{x \lambda }}{\lambda }\right )+c_1} \\ y(x)\to a e^{\lambda x} \\ \end{align*}