61.3.9 problem 9

Internal problem ID [12093]
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
Date solved : Tuesday, January 28, 2025 at 12:21:59 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=a \,{\mathrm e}^{k x} y^{2}+b \,{\mathrm e}^{s x} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 228

dsolve(diff(y(x),x)=a*exp(k*x)*y(x)^2+b*exp(s*x),y(x), singsol=all)
 

\[ y = -\frac {b \,{\mathrm e}^{s x} \left (\operatorname {BesselY}\left (\frac {s}{s +k}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {s}{s +k}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right )\right )}{\operatorname {BesselJ}\left (\frac {k +2 s}{s +k}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right ) \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}+\sqrt {b}\, \sqrt {a}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}} \operatorname {BesselY}\left (\frac {k +2 s}{s +k}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right ) c_{1} -s \left (\operatorname {BesselY}\left (\frac {s}{s +k}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {s}{s +k}, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right )\right )} \]

✓ Solution by Mathematica

Time used: 2.631 (sec). Leaf size: 859

DSolve[D[y[x],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}} \operatorname {BesselI}\left (\frac {k \log \left (e^{k+s}\right )}{(k+s)^2},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 (\operatorname {BesselI}\left (\frac {k \log \left (e^{k+s}\right )-(k+s)^2}{(k+s)^2},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 )+\operatorname {BesselI}\left (\frac {k \log \left (e^{k+s}\right )}{(k+s)^2}+1,2 \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}} \operatorname {BesselI}\left (\frac {k \log \left (e^{k+s}\right )}{(k+s)^2},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 (\operatorname {BesselI}\left (\frac {k \log \left (e^{k+s}\right )-(k+s)^2}{(k+s)^2},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 )+\operatorname {BesselI}\left (\frac {k \log \left (e^{k+s}\right )}{(k+s)^2}+1,2 \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 )}{\operatorname {BesselI}\left (\frac {k \log \left (e^{k+s}\right )}{(k+s)^2},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*}