Internal problem ID [10411]
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: 3.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-\sigma y^{2}=a +b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \lambda x}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 348
dsolve(diff(y(x),x)=sigma*y(x)^2+a+b*exp(lambda*x)+c*exp(2*lambda*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {-\operatorname {WhittakerM}\left (-\frac {i \sqrt {\sigma }\, b -2 \lambda \sqrt {c}}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{x \lambda }}{\lambda }\right ) \left (i \left (\sqrt {c}\, \sqrt {a}-\frac {b}{2}\right ) \sqrt {\sigma }+\frac {\lambda \sqrt {c}}{2}\right )+\lambda c_{1} \operatorname {WhittakerW}\left (-\frac {i \sqrt {\sigma }\, b -2 \lambda \sqrt {c}}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{x \lambda }}{\lambda }\right ) \sqrt {c}+\left (-i {\mathrm e}^{x \lambda } c \sqrt {\sigma }-\frac {i \sqrt {\sigma }\, b}{2}+\frac {\lambda \sqrt {c}}{2}\right ) \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {\sigma }\, b}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{x \lambda }}{\lambda }\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {i \sqrt {\sigma }\, b}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{x \lambda }}{\lambda }\right )\right )}{\sqrt {c}\, \sigma \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {\sigma }\, b}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{x \lambda }}{\lambda }\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {i \sqrt {\sigma }\, b}{2 \lambda \sqrt {c}}, \frac {i \sqrt {a}\, \sqrt {\sigma }}{\lambda }, \frac {2 i \sqrt {\sigma }\, \sqrt {c}\, {\mathrm e}^{x \lambda }}{\lambda }\right )\right )} \]
✓ Solution by Mathematica
Time used: 3.251 (sec). Leaf size: 1081
DSolve[y'[x]==sigma*y[x]^2+a+b*Exp[\[Lambda]*x]+c*Exp[2*\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {i \left (c_1 \lambda \left (\sqrt {a}-\sqrt {c} e^{\lambda x}\right ) \operatorname {HypergeometricU}\left (\frac {\frac {i \sqrt {\sigma } b}{\sqrt {c}}+\lambda +2 i \sqrt {a} \sqrt {\sigma }}{2 \lambda },\frac {2 i \sqrt {a} \sqrt {\sigma }}{\lambda }+1,\frac {2 i \sqrt {c} e^{x \lambda } \sqrt {\sigma }}{\lambda }\right )-i c_1 e^{\lambda x} \left (b \sqrt {\sigma }+\sqrt {c} \left (2 \sqrt {a} \sqrt {\sigma }-i \lambda \right )\right ) \operatorname {HypergeometricU}\left (\frac {\frac {i \sqrt {\sigma } b}{\sqrt {c}}+3 \lambda +2 i \sqrt {a} \sqrt {\sigma }}{2 \lambda },\frac {2 i \sqrt {a} \sqrt {\sigma }}{\lambda }+2,\frac {2 i \sqrt {c} e^{x \lambda } \sqrt {\sigma }}{\lambda }\right )+\lambda \left (\left (\sqrt {a}-\sqrt {c} e^{\lambda x}\right ) L_{-\frac {\frac {i \sqrt {\sigma } b}{\sqrt {c}}+\lambda +2 i \sqrt {a} \sqrt {\sigma }}{2 \lambda }}^{\frac {2 i \sqrt {a} \sqrt {\sigma }}{\lambda }}\left (\frac {2 i \sqrt {c} e^{x \lambda } \sqrt {\sigma }}{\lambda }\right )-2 \sqrt {c} e^{\lambda x} L_{-\frac {\frac {i \sqrt {\sigma } b}{\sqrt {c}}+3 \lambda +2 i \sqrt {a} \sqrt {\sigma }}{2 \lambda }}^{\frac {2 i \sqrt {a} \sqrt {\sigma }}{\lambda }+1}\left (\frac {2 i \sqrt {c} e^{x \lambda } \sqrt {\sigma }}{\lambda }\right )\right )\right )}{\lambda \sqrt {\sigma } \left (c_1 \operatorname {HypergeometricU}\left (\frac {\frac {i \sqrt {\sigma } b}{\sqrt {c}}+\lambda +2 i \sqrt {a} \sqrt {\sigma }}{2 \lambda },\frac {2 i \sqrt {a} \sqrt {\sigma }}{\lambda }+1,\frac {2 i \sqrt {c} e^{x \lambda } \sqrt {\sigma }}{\lambda }\right )+L_{-\frac {\frac {i \sqrt {\sigma } b}{\sqrt {c}}+\lambda +2 i \sqrt {a} \sqrt {\sigma }}{2 \lambda }}^{\frac {2 i \sqrt {a} \sqrt {\sigma }}{\lambda }}\left (\frac {2 i \sqrt {c} e^{x \lambda } \sqrt {\sigma }}{\lambda }\right )\right )} \\ y(x)\to \frac {-\frac {e^{\lambda x} \left (b \sqrt {\sigma }+\sqrt {c} \left (2 \sqrt {a} \sqrt {\sigma }-i \lambda \right )\right ) \operatorname {HypergeometricU}\left (\frac {\frac {i \sqrt {\sigma } b}{\sqrt {c}}+3 \lambda +2 i \sqrt {a} \sqrt {\sigma }}{2 \lambda },\frac {2 i \sqrt {a} \sqrt {\sigma }}{\lambda }+2,\frac {2 i \sqrt {c} e^{x \lambda } \sqrt {\sigma }}{\lambda }\right )}{\lambda \operatorname {HypergeometricU}\left (\frac {\frac {i \sqrt {\sigma } b}{\sqrt {c}}+\lambda +2 i \sqrt {a} \sqrt {\sigma }}{2 \lambda },\frac {2 i \sqrt {a} \sqrt {\sigma }}{\lambda }+1,\frac {2 i \sqrt {c} e^{x \lambda } \sqrt {\sigma }}{\lambda }\right )}-i \left (\sqrt {a}-\sqrt {c} e^{\lambda x}\right )}{\sqrt {\sigma }} \\ y(x)\to \frac {-\frac {e^{\lambda x} \left (b \sqrt {\sigma }+\sqrt {c} \left (2 \sqrt {a} \sqrt {\sigma }-i \lambda \right )\right ) \operatorname {HypergeometricU}\left (\frac {\frac {i \sqrt {\sigma } b}{\sqrt {c}}+3 \lambda +2 i \sqrt {a} \sqrt {\sigma }}{2 \lambda },\frac {2 i \sqrt {a} \sqrt {\sigma }}{\lambda }+2,\frac {2 i \sqrt {c} e^{x \lambda } \sqrt {\sigma }}{\lambda }\right )}{\lambda \operatorname {HypergeometricU}\left (\frac {\frac {i \sqrt {\sigma } b}{\sqrt {c}}+\lambda +2 i \sqrt {a} \sqrt {\sigma }}{2 \lambda },\frac {2 i \sqrt {a} \sqrt {\sigma }}{\lambda }+1,\frac {2 i \sqrt {c} e^{x \lambda } \sqrt {\sigma }}{\lambda }\right )}-i \left (\sqrt {a}-\sqrt {c} e^{\lambda x}\right )}{\sqrt {\sigma }} \\ \end{align*}