Internal problem ID [9665]
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: 4.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-\sigma y^{2}-y a -b \,{\mathrm e}^{x}-c=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 317
dsolve(diff(y(x),x)=sigma*y(x)^2+a*y(x)+b*exp(x)+c,y(x), singsol=all)
\[ y \left (x \right ) = \left (\frac {\sqrt {b}\, c_{1} \operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }+1, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right )}{\sqrt {\sigma }\, \left (\operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right ) c_{1} +\operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right )\right )}+\frac {\operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }+1, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right ) \sqrt {b}}{\sqrt {\sigma }\, \left (\operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right ) c_{1} +\operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right )\right )}\right ) {\mathrm e}^{\frac {x}{2}}+\frac {\left (-\sqrt {a^{2}-4 c \sigma }\, c_{1} -c_{1} a \right ) \operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right )+\left (-\sqrt {a^{2}-4 c \sigma }-a \right ) \operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right )}{2 \sigma \left (\operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right ) c_{1} +\operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.596 (sec). Leaf size: 482
DSolve[y'[x]==sigma*y[x]^2+a*y[x]+b*Exp[x]+c,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {b \sigma e^x} \left (\operatorname {Gamma}\left (\sqrt {a^2-4 c \sigma }+1\right ) \left (2 \operatorname {BesselJ}\left (\sqrt {a^2-4 c \sigma }+1,2 \sqrt {b e^x \sigma }\right )-\frac {\left (\sqrt {a^2-4 c \sigma }+a\right ) \operatorname {BesselJ}\left (\sqrt {a^2-4 c \sigma },2 \sqrt {b e^x \sigma }\right )}{\sqrt {b \sigma e^x}}\right )-\frac {c_1 \left (\sqrt {a^2-4 c \sigma }+a\right ) \operatorname {Gamma}\left (1-\sqrt {a^2-4 c \sigma }\right ) \operatorname {BesselJ}\left (-\sqrt {a^2-4 c \sigma },2 \sqrt {b e^x \sigma }\right )}{\sqrt {b \sigma e^x}}-2 c_1 \operatorname {Gamma}\left (1-\sqrt {a^2-4 c \sigma }\right ) \operatorname {BesselJ}\left (-\sqrt {a^2-4 c \sigma }-1,2 \sqrt {b e^x \sigma }\right )\right )}{2 \left (\sigma \operatorname {Gamma}\left (\sqrt {a^2-4 c \sigma }+1\right ) \operatorname {BesselJ}\left (\sqrt {a^2-4 c \sigma },2 \sqrt {b e^x \sigma }\right )+c_1 \sigma \operatorname {Gamma}\left (1-\sqrt {a^2-4 c \sigma }\right ) \operatorname {BesselJ}\left (-\sqrt {a^2-4 c \sigma },2 \sqrt {b e^x \sigma }\right )\right )} \\ y(x)\to -\frac {\frac {2 \, _0\tilde {F}_1\left (;-\sqrt {a^2-4 c \sigma };-b e^x \sigma \right )}{\, _0\tilde {F}_1\left (;1-\sqrt {a^2-4 c \sigma };-b e^x \sigma \right )}+\sqrt {a^2-4 c \sigma }+a}{2 \sigma } \\ y(x)\to -\frac {\frac {2 \, _0\tilde {F}_1\left (;-\sqrt {a^2-4 c \sigma };-b e^x \sigma \right )}{\, _0\tilde {F}_1\left (;1-\sqrt {a^2-4 c \sigma };-b e^x \sigma \right )}+\sqrt {a^2-4 c \sigma }+a}{2 \sigma } \\ \end{align*}