Internal problem ID [10421]
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: 13.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-{\mathrm e}^{\lambda x} y^{2}-a \,{\mathrm e}^{x \mu } y=a \lambda \,{\mathrm e}^{\left (\mu -\lambda \right ) x}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 97
dsolve(diff(y(x),x)=exp(lambda*x)*y(x)^2+a*exp(mu*x)*y(x)+a*lambda*exp((mu-lambda)*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\lambda \left (a c_{1} {\mathrm e}^{\left (\mu -\lambda \right ) x} \operatorname {hypergeom}\left (\left [\frac {\mu -\lambda }{\mu }\right ], \left [\frac {-\lambda +2 \mu }{\mu }\right ], \frac {a \,{\mathrm e}^{x \mu }}{\mu }\right )+\lambda -\mu \right )}{\left (\mu -\lambda \right ) \left (c_{1} \operatorname {hypergeom}\left (\left [-\frac {\lambda }{\mu }\right ], \left [\frac {\mu -\lambda }{\mu }\right ], \frac {a \,{\mathrm e}^{x \mu }}{\mu }\right )+{\mathrm e}^{x \lambda }\right )} \]
✓ Solution by Mathematica
Time used: 4.392 (sec). Leaf size: 148
DSolve[y'[x]==Exp[\[Lambda]*x]*y[x]^2+a*Exp[\[Mu]*x]*y[x]+a*\[Lambda]*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 )}{-\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 }} \\ y(x)\to \lambda \left (-e^{\lambda (-x)}\right ) \\ \end{align*}