Internal problem ID [10419]
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: 1.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-a y^{2}=b \,{\mathrm e}^{\lambda x}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 144
dsolve(diff(y(x),x)=a*y(x)^2+b*exp(lambda*x),y(x), singsol=all)
\[ y \left (x \right ) = \left (\frac {\sqrt {b}\, c_{1} \operatorname {BesselY}\left (1, \frac {2 \sqrt {b}\, \sqrt {a}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )}{\sqrt {a}\, \left (c_{1} \operatorname {BesselY}\left (0, \frac {2 \sqrt {b}\, \sqrt {a}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )+\operatorname {BesselJ}\left (0, \frac {2 \sqrt {b}\, \sqrt {a}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )\right )}+\frac {\sqrt {b}\, \operatorname {BesselJ}\left (1, \frac {2 \sqrt {b}\, \sqrt {a}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )}{\sqrt {a}\, \left (c_{1} \operatorname {BesselY}\left (0, \frac {2 \sqrt {b}\, \sqrt {a}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )+\operatorname {BesselJ}\left (0, \frac {2 \sqrt {b}\, \sqrt {a}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )\right )}\right ) {\mathrm e}^{\frac {\lambda x}{2}} \]
✓ Solution by Mathematica
Time used: 0.551 (sec). Leaf size: 266
DSolve[y'[x]==a*y[x]^2+b*Exp[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {b e^{\lambda x}} \left (2 \operatorname {BesselY}\left (1,\frac {2 \sqrt {a} \sqrt {b e^{x \lambda }}}{\lambda }\right )+c_1 \operatorname {BesselJ}\left (1,\frac {2 \sqrt {a} \sqrt {b e^{x \lambda }}}{\lambda }\right )\right )}{\sqrt {a} \left (2 \operatorname {BesselY}\left (0,\frac {2 \sqrt {a} \sqrt {b e^{x \lambda }}}{\lambda }\right )+c_1 \operatorname {BesselJ}\left (0,\frac {2 \sqrt {a} \sqrt {b e^{x \lambda }}}{\lambda }\right )\right )} y(x)\to \frac {\sqrt {b e^{\lambda x}} \operatorname {BesselJ}\left (1,\frac {2 \sqrt {a} \sqrt {b e^{x \lambda }}}{\lambda }\right )}{\sqrt {a} \operatorname {BesselJ}\left (0,\frac {2 \sqrt {a} \sqrt {b e^{x \lambda }}}{\lambda }\right )} y(x)\to \frac {\sqrt {b e^{\lambda x}} \operatorname {BesselJ}\left (1,\frac {2 \sqrt {a} \sqrt {b e^{x \lambda }}}{\lambda }\right )}{\sqrt {a} \operatorname {BesselJ}\left (0,\frac {2 \sqrt {a} \sqrt {b e^{x \lambda }}}{\lambda }\right )} \end{align*}