Internal problem ID [9679]
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: 14.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }+\lambda \,{\mathrm e}^{\lambda x} y^{2}-a \,{\mathrm e}^{\mu x} y+a \,{\mathrm e}^{\left (-\lambda +\mu \right ) x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 136
dsolve(diff(y(x),x)=-lambda*exp(lambda*x)*y(x)^2+a*exp(mu*x)*y(x)-a*exp((mu-lambda)*x),y(x), singsol=all)
\[ y \relax (x ) = \left (\frac {\hypergeom \left (\left [-\frac {\lambda -\mu }{\mu }\right ], \left [-\frac {\lambda -2 \mu }{\mu }\right ], \frac {{\mathrm e}^{\mu x} a}{\mu }\right ) c_{1} a \,{\mathrm e}^{\mu x}}{\left (\lambda -\mu \right ) \left (c_{1} \hypergeom \left (\left [-\frac {\lambda }{\mu }\right ], \left [-\frac {\lambda -\mu }{\mu }\right ], \frac {{\mathrm e}^{\mu x} a}{\mu }\right )+{\mathrm e}^{\lambda x}\right )}+\frac {{\mathrm e}^{\lambda x}}{c_{1} \hypergeom \left (\left [-\frac {\lambda }{\mu }\right ], \left [-\frac {\lambda -\mu }{\mu }\right ], \frac {{\mathrm e}^{\mu x} a}{\mu }\right )+{\mathrm e}^{\lambda x}}\right ) {\mathrm e}^{-\lambda x} \]
✓ Solution by Mathematica
Time used: 3.157 (sec). Leaf size: 78
DSolve[y'[x]==-\[Lambda]*Exp[\[Lambda]*x]*y[x]^2+a*Exp[\[Mu]*x]*y[x]-a*Exp[(\[Mu]-\[Lambda])*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{\lambda (-x)} \left (1+\frac {\mu e^{\frac {a e^{\mu x}}{\mu }}}{-\lambda E_{\frac {\lambda +\mu }{\mu }}\left (-\frac {a e^{x \mu }}{\mu }\right )+c_1 \lambda \left (e^{\mu x}\right )^{\lambda /\mu }}\right ) \\ y(x)\to e^{\lambda (-x)} \\ \end{align*}