Internal problem ID [9674]
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: 9.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-a \,{\mathrm e}^{k x} y^{2}-b \,{\mathrm e}^{s x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 145
dsolve(diff(y(x),x)=a*exp(k*x)*y(x)^2+b*exp(s*x),y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (\BesselY \left (\frac {s}{s +k}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right ) c_{1}+\BesselJ \left (\frac {s}{s +k}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right )\right ) \sqrt {b}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}-k x}}{\sqrt {a}\, \left (\BesselY \left (-\frac {k}{s +k}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right ) c_{1}+\BesselJ \left (-\frac {k}{s +k}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right )\right )} \]
✓ Solution by Mathematica
Time used: 5.696 (sec). Leaf size: 1104
DSolve[y'[x]==a*Exp[k*x]*y[x]^2+b*Exp[s*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-k x} \left (-k K_{\frac {k \log \left (e^{k+s}\right )}{(k+s)^2}}\left (2 \sqrt {-\frac {a b \left (\left (e^{k+s}\right )^x\right )^{\frac {k+s}{\log \left (e^{k+s}\right )}} \log ^2\left (e^{k+s}\right )}{(k+s)^4}}\right )-c_1 k (-1)^{\frac {k \log \left (e^{k+s}\right )}{(k+s)^2}} I_{\frac {k \log \left (e^{k+s}\right )}{(k+s)^2}}\left (2 \sqrt {-\frac {a b \left (\left (e^{k+s}\right )^x\right )^{\frac {k+s}{\log \left (e^{k+s}\right )}} \log ^2\left (e^{k+s}\right )}{(k+s)^4}}\right )+(k+s) \sqrt {-\frac {a b \log ^2\left (e^{k+s}\right ) \left (\left (e^{k+s}\right )^x\right )^{\frac {k+s}{\log \left (e^{k+s}\right )}}}{(k+s)^4}} \left (K_{\frac {k \log \left (e^{k+s}\right )-(k+s)^2}{(k+s)^2}}\left (2 \sqrt {-\frac {a b \left (\left (e^{k+s}\right )^x\right )^{\frac {k+s}{\log \left (e^{k+s}\right )}} \log ^2\left (e^{k+s}\right )}{(k+s)^4}}\right )+K_{\frac {k \log \left (e^{k+s}\right )}{(k+s)^2}+1}\left (2 \sqrt {-\frac {a b \left (\left (e^{k+s}\right )^x\right )^{\frac {k+s}{\log \left (e^{k+s}\right )}} \log ^2\left (e^{k+s}\right )}{(k+s)^4}}\right )-c_1 (-1)^{\frac {k \log \left (e^{k+s}\right )}{(k+s)^2}} \left (I_{\frac {k \log \left (e^{k+s}\right )-(k+s)^2}{(k+s)^2}}\left (2 \sqrt {-\frac {a b \left (\left (e^{k+s}\right )^x\right )^{\frac {k+s}{\log \left (e^{k+s}\right )}} \log ^2\left (e^{k+s}\right )}{(k+s)^4}}\right )+I_{\frac {k \log \left (e^{k+s}\right )}{(k+s)^2}+1}\left (2 \sqrt {-\frac {a b \left (\left (e^{k+s}\right )^x\right )^{\frac {k+s}{\log \left (e^{k+s}\right )}} \log ^2\left (e^{k+s}\right )}{(k+s)^4}}\right )\right )\right )\right )}{2 a \left (K_{\frac {k \log \left (e^{k+s}\right )}{(k+s)^2}}\left (2 \sqrt {-\frac {a b \left (\left (e^{k+s}\right )^x\right )^{\frac {k+s}{\log \left (e^{k+s}\right )}} \log ^2\left (e^{k+s}\right )}{(k+s)^4}}\right )+c_1 (-1)^{\frac {k \log \left (e^{k+s}\right )}{(k+s)^2}} I_{\frac {k \log \left (e^{k+s}\right )}{(k+s)^2}}\left (2 \sqrt {-\frac {a b \left (\left (e^{k+s}\right )^x\right )^{\frac {k+s}{\log \left (e^{k+s}\right )}} \log ^2\left (e^{k+s}\right )}{(k+s)^4}}\right )\right )} \\ y(x)\to \frac {e^{-k x} \left (-\frac {(k+s) \sqrt {-\frac {a b \log ^2\left (e^{k+s}\right ) \left (\left (e^{k+s}\right )^x\right )^{\frac {k+s}{\log \left (e^{k+s}\right )}}}{(k+s)^4}} \left (I_{\frac {k \log \left (e^{k+s}\right )}{(k+s)^2}-1}\left (2 \sqrt {-\frac {a b \left (\left (e^{k+s}\right )^x\right )^{\frac {k+s}{\log \left (e^{k+s}\right )}} \log ^2\left (e^{k+s}\right )}{(k+s)^4}}\right )+I_{\frac {k \log \left (e^{k+s}\right )}{(k+s)^2}+1}\left (2 \sqrt {-\frac {a b \left (\left (e^{k+s}\right )^x\right )^{\frac {k+s}{\log \left (e^{k+s}\right )}} \log ^2\left (e^{k+s}\right )}{(k+s)^4}}\right )\right )}{I_{\frac {k \log \left (e^{k+s}\right )}{(k+s)^2}}\left (2 \sqrt {-\frac {a b \left (\left (e^{k+s}\right )^x\right )^{\frac {k+s}{\log \left (e^{k+s}\right )}} \log ^2\left (e^{k+s}\right )}{(k+s)^4}}\right )}-k\right )}{2 a} \\ y(x)\to \frac {e^{-k x} \left (-\frac {(k+s) \sqrt {-\frac {a b \log ^2\left (e^{k+s}\right ) \left (\left (e^{k+s}\right )^x\right )^{\frac {k+s}{\log \left (e^{k+s}\right )}}}{(k+s)^4}} \left (I_{\frac {k \log \left (e^{k+s}\right )-(k+s)^2}{(k+s)^2}}\left (2 \sqrt {-\frac {a b \left (\left (e^{k+s}\right )^x\right )^{\frac {k+s}{\log \left (e^{k+s}\right )}} \log ^2\left (e^{k+s}\right )}{(k+s)^4}}\right )+I_{\frac {k \log \left (e^{k+s}\right )}{(k+s)^2}+1}\left (2 \sqrt {-\frac {a b \left (\left (e^{k+s}\right )^x\right )^{\frac {k+s}{\log \left (e^{k+s}\right )}} \log ^2\left (e^{k+s}\right )}{(k+s)^4}}\right )\right )}{I_{\frac {k \log \left (e^{k+s}\right )}{(k+s)^2}}\left (2 \sqrt {-\frac {a b \left (\left (e^{k+s}\right )^x\right )^{\frac {k+s}{\log \left (e^{k+s}\right )}} \log ^2\left (e^{k+s}\right )}{(k+s)^4}}\right )}-k\right )}{2 a} \\ \end{align*}