Internal problem ID [10422]
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]
\[ \boxed {y^{\prime }+{\mathrm e}^{\lambda x} y^{2} \lambda -a \,{\mathrm e}^{x \mu } y=-a \,{\mathrm e}^{\left (\mu -\lambda \right ) x}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 96
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)
\[ y \left (x \right ) = \frac {a c_{1} {\mathrm e}^{\left (\mu -\lambda \right ) x} \operatorname {hypergeom}\left (\left [\frac {\mu -\lambda }{\mu }\right ], \left [\frac {-\lambda +2 \mu }{\mu }\right ], \frac {a \,{\mathrm e}^{x \mu }}{\mu }\right )+\lambda -\mu }{\left (\lambda -\mu \right ) \left (c_{1} \operatorname {hypergeom}\left (\left [-\frac {\lambda }{\mu }\right ], \left [\frac {\mu -\lambda }{\mu }\right ], \frac {a \,{\mathrm e}^{x \mu }}{\mu }\right )+{\mathrm e}^{x \lambda }\right )} \]
✓ Solution by Mathematica
Time used: 4.358 (sec). Leaf size: 147
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 \frac {e^{\lambda (-x)} \left (-\lambda \left (-\frac {a e^{\mu x}}{\mu }\right )^{\lambda /\mu } \Gamma \left (-\frac {\lambda }{\mu },-\frac {a e^{x \mu }}{\mu }\right )+\mu e^{\frac {a e^{\mu x}}{\mu }}+c_1 \lambda \left (e^{\mu x}\right )^{\lambda /\mu }\right )}{\lambda \left (-\left (-\frac {a e^{\mu x}}{\mu }\right )^{\lambda /\mu } \Gamma \left (-\frac {\lambda }{\mu },-\frac {a e^{x \mu }}{\mu }\right )+c_1 \left (e^{\mu x}\right )^{\lambda /\mu }\right )} \\ y(x)\to e^{\lambda (-x)} \\ \end{align*}