Internal problem ID [9666]
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]
Solve \begin {gather*} \boxed {y^{\prime }-a y^{2}-b \,{\mathrm e}^{\lambda x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 144
dsolve(diff(y(x),x)=a*y(x)^2+b*exp(lambda*x),y(x), singsol=all)
\[ y \relax (x ) = \left (\frac {\sqrt {b}\, c_{1} \BesselY \left (1, \frac {2 \sqrt {b}\, \sqrt {a}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )}{\sqrt {a}\, \left (c_{1} \BesselY \left (0, \frac {2 \sqrt {b}\, \sqrt {a}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )+\BesselJ \left (0, \frac {2 \sqrt {b}\, \sqrt {a}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )\right )}+\frac {\sqrt {b}\, \BesselJ \left (1, \frac {2 \sqrt {b}\, \sqrt {a}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )}{\sqrt {a}\, \left (c_{1} \BesselY \left (0, \frac {2 \sqrt {b}\, \sqrt {a}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )+\BesselJ \left (0, \frac {2 \sqrt {b}\, \sqrt {a}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )\right )}\right ) {\mathrm e}^{\frac {\lambda x}{2}} \]
✓ Solution by Mathematica
Time used: 0.297 (sec). Leaf size: 163
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 Y_1\left (\frac {2 \sqrt {a} \sqrt {b e^{x \lambda }}}{\lambda }\right )+c_1 J_1\left (\frac {2 \sqrt {a} \sqrt {b e^{x \lambda }}}{\lambda }\right )\right )}{\sqrt {a} \left (2 Y_0\left (\frac {2 \sqrt {a} \sqrt {b e^{x \lambda }}}{\lambda }\right )+c_1 \, _0\tilde {F}_1\left (;1;-\frac {a b e^{x \lambda }}{\lambda ^2}\right )\right )} \\ y(x)\to \frac {b e^{\lambda x} \, _0\tilde {F}_1\left (;2;-\frac {a b e^{x \lambda }}{\lambda ^2}\right )}{\lambda \, _0\tilde {F}_1\left (;1;-\frac {a b e^{x \lambda }}{\lambda ^2}\right )} \\ \end{align*}