Internal problem ID [10431]
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]
\[ \boxed {y^{\prime }-y^{2} {\mathrm e}^{\lambda x}-a \,{\mathrm e}^{\mu x} y=a \lambda \,{\mathrm e}^{\left (\mu -\lambda \right ) x}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 140
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)
\[ y \left (x \right ) = \left (-\frac {\lambda \operatorname {hypergeom}\left (\left [-\frac {\lambda -\mu }{\mu }\right ], \left [-\frac {\lambda -2 \mu }{\mu }\right ], \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right ) c_{1} a \,{\mathrm e}^{\mu x}}{\left (\lambda -\mu \right ) \left (c_{1} \operatorname {hypergeom}\left (\left [-\frac {\lambda }{\mu }\right ], \left [-\frac {\lambda -\mu }{\mu }\right ], \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right )+{\mathrm e}^{\lambda x}\right )}-\frac {\lambda \,{\mathrm e}^{\lambda x}}{c_{1} \operatorname {hypergeom}\left (\left [-\frac {\lambda }{\mu }\right ], \left [-\frac {\lambda -\mu }{\mu }\right ], \frac {a \,{\mathrm e}^{\mu x}}{\mu }\right )+{\mathrm e}^{\lambda x}}\right ) {\mathrm e}^{-\lambda x} \]
✓ 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*}