Internal problem ID [9675]
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: 10.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-b \,{\mathrm e}^{\mu x} y^{2}-a \lambda \,{\mathrm e}^{\lambda x}+a^{2} b \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x}=0} \end {gather*}
✗ Solution by Maple
dsolve(diff(y(x),x)=b*exp(mu*x)*y(x)^2+a*lambda*exp(lambda*x)-a^2*b*exp((mu+2*lambda)*x),y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 4.2 (sec). Leaf size: 844
DSolve[y'[x]==b*Exp[\[Mu]*x]*y[x]^2+a*\[Lambda]*Exp[\[Lambda]*x]-a^2*b*Exp[(\[Mu]+2*\[Lambda])*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{\mu (-x)} \left (-2 a b \log \left (e^{\lambda +\mu }\right ) \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}} \left (2 (\lambda +\mu ) L_{-\frac {\log \left (e^{\lambda +\mu }\right )}{2 (\lambda +\mu )}-\frac {3}{2}}^{\frac {\mu \log \left (e^{\lambda +\mu }\right )}{(\lambda +\mu )^2}+1}\left (-\frac {2 a b \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}} \log \left (e^{\lambda +\mu }\right )}{(\lambda +\mu )^2}\right )+c_1 \left (\log \left (e^{\lambda +\mu }\right )+\lambda +\mu \right ) \text {HypergeometricU}\left (\frac {1}{2} \left (\frac {\log \left (e^{\lambda +\mu }\right )}{\lambda +\mu }+3\right ),\frac {2 (\lambda +\mu )^2+\mu \log \left (e^{\lambda +\mu }\right )}{(\lambda +\mu )^2},-\frac {2 a b \log \left (e^{\lambda +\mu }\right ) \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}}}{(\lambda +\mu )^2}\right )\right )-c_1 (\lambda +\mu ) \left (\log \left (e^{\lambda +\mu }\right ) \left (2 a b \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}}+\mu \right )+\mu (\lambda +\mu )\right ) \text {HypergeometricU}\left (\frac {\log \left (e^{\lambda +\mu }\right )+\lambda +\mu }{2 (\lambda +\mu )},\frac {\mu \log \left (e^{\lambda +\mu }\right )}{(\lambda +\mu )^2}+1,-\frac {2 a b \log \left (e^{\lambda +\mu }\right ) \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}}}{(\lambda +\mu )^2}\right )-(\lambda +\mu ) \left (\log \left (e^{\lambda +\mu }\right ) \left (2 a b \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}}+\mu \right )+\mu (\lambda +\mu )\right ) L_{-\frac {\lambda +\mu +\log \left (e^{\lambda +\mu }\right )}{2 (\lambda +\mu )}}^{\frac {\mu \log \left (e^{\lambda +\mu }\right )}{(\lambda +\mu )^2}}\left (-\frac {2 a b \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}} \log \left (e^{\lambda +\mu }\right )}{(\lambda +\mu )^2}\right )\right )}{2 b (\lambda +\mu )^2 \left (L_{-\frac {\lambda +\mu +\log \left (e^{\lambda +\mu }\right )}{2 (\lambda +\mu )}}^{\frac {\mu \log \left (e^{\lambda +\mu }\right )}{(\lambda +\mu )^2}}\left (-\frac {2 a b \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}} \log \left (e^{\lambda +\mu }\right )}{(\lambda +\mu )^2}\right )+c_1 \text {HypergeometricU}\left (\frac {\log \left (e^{\lambda +\mu }\right )+\lambda +\mu }{2 (\lambda +\mu )},\frac {\mu \log \left (e^{\lambda +\mu }\right )}{(\lambda +\mu )^2}+1,-\frac {2 a b \log \left (e^{\lambda +\mu }\right ) \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}}}{(\lambda +\mu )^2}\right )\right )} \\ y(x)\to -\frac {a e^{\mu (-x)} \log \left (e^{\lambda +\mu }\right ) \left (\log \left (e^{\lambda +\mu }\right )+\lambda +\mu \right ) \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}} \text {HypergeometricU}\left (\frac {1}{2} \left (\frac {\log \left (e^{\lambda +\mu }\right )}{\lambda +\mu }+3\right ),\frac {2 (\lambda +\mu )^2+\mu \log \left (e^{\lambda +\mu }\right )}{(\lambda +\mu )^2},-\frac {2 a b \log \left (e^{\lambda +\mu }\right ) \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}}}{(\lambda +\mu )^2}\right )}{(\lambda +\mu )^2 \text {HypergeometricU}\left (\frac {\log \left (e^{\lambda +\mu }\right )+\lambda +\mu }{2 (\lambda +\mu )},\frac {\mu \log \left (e^{\lambda +\mu }\right )}{(\lambda +\mu )^2}+1,-\frac {2 a b \log \left (e^{\lambda +\mu }\right ) \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}}}{(\lambda +\mu )^2}\right )}-\frac {e^{\mu (-x)} \left (\log \left (e^{\lambda +\mu }\right ) \left (2 a b \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}}+\mu \right )+\mu (\lambda +\mu )\right )}{2 b (\lambda +\mu )} \\ \end{align*}