Internal problem ID [10434]
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: 16.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-a \,{\mathrm e}^{k x} y^{2}-b y=c \,{\mathrm e}^{s x}+d \,{\mathrm e}^{-k x}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 563
dsolve(diff(y(x),x)=a*exp(k*x)*y(x)^2+b*y(x)+c*exp(s*x)+d*exp(-k*x),y(x), singsol=all)
\[ y \left (x \right ) = \left (\frac {\sqrt {c}\, c_{1} \operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}+s +k}{s +k}, \frac {2 \sqrt {a}\, \sqrt {c}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right )}{\sqrt {a}\, \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}}{s +k}, \frac {2 \sqrt {a}\, \sqrt {c}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}}{s +k}, \frac {2 \sqrt {a}\, \sqrt {c}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right )\right )}+\frac {\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}+s +k}{s +k}, \frac {2 \sqrt {a}\, \sqrt {c}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right ) \sqrt {c}}{\sqrt {a}\, \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}}{s +k}, \frac {2 \sqrt {a}\, \sqrt {c}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}}{s +k}, \frac {2 \sqrt {a}\, \sqrt {c}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right )\right )}\right ) {\mathrm e}^{-x k} {\mathrm e}^{\frac {x \left (s +k \right )}{2}}+\frac {\left (\left (-\sqrt {-4 a d +b^{2}+2 b k +k^{2}}\, c_{1} -c_{1} b -c_{1} k \right ) \operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}}{s +k}, \frac {2 \sqrt {a}\, \sqrt {c}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right )+\left (-\sqrt {-4 a d +b^{2}+2 b k +k^{2}}-b -k \right ) \operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}}{s +k}, \frac {2 \sqrt {a}\, \sqrt {c}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right )\right ) {\mathrm e}^{-x k}}{2 \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}}{s +k}, \frac {2 \sqrt {a}\, \sqrt {c}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}}{s +k}, \frac {2 \sqrt {a}\, \sqrt {c}\, {\mathrm e}^{\frac {x \left (s +k \right )}{2}}}{s +k}\right )\right ) a} \]
✓ Solution by Mathematica
Time used: 18.386 (sec). Leaf size: 1636
DSolve[y'[x]==a*Exp[k*x]*y[x]^2+b*y[x]+c*Exp[s*x]+d*Exp[-k*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-k x} \left (-\left ((b+k) K_{\frac {\sqrt {\left (b^2+2 k b+k^2-4 a d\right ) (k+s)^4 \log ^2\left (e^{k+s}\right )}}{(k+s)^4}}\left (2 \sqrt {-\frac {a c \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 )+(-1)^{\frac {k^4+4 s k^3+6 s^2 k^2+4 s^3 k+s^4+\sqrt {\left (b^2+2 k b+k^2-4 a d\right ) (k+s)^4 \log ^2\left (e^{k+s}\right )}}{(k+s)^4}} (b+k) \operatorname {BesselI}\left (\frac {\sqrt {\left (b^2+2 k b+k^2-4 a d\right ) (k+s)^4 \log ^2\left (e^{k+s}\right )}}{(k+s)^4},2 \sqrt {-\frac {a c \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+s) \left (K_{-\frac {k^4+4 s k^3+6 s^2 k^2+4 s^3 k+s^4-\sqrt {\left (b^2+2 k b+k^2-4 a d\right ) (k+s)^4 \log ^2\left (e^{k+s}\right )}}{(k+s)^4}}\left (2 \sqrt {-\frac {a c \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^4+4 s k^3+6 s^2 k^2+4 s^3 k+s^4+\sqrt {\left (b^2+2 k b+k^2-4 a d\right ) (k+s)^4 \log ^2\left (e^{k+s}\right )}}{(k+s)^4}}\left (2 \sqrt {-\frac {a c \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 )+(-1)^{\frac {k^4+4 s k^3+6 s^2 k^2+4 s^3 k+s^4+\sqrt {\left (b^2+2 k b+k^2-4 a d\right ) (k+s)^4 \log ^2\left (e^{k+s}\right )}}{(k+s)^4}} \left (\operatorname {BesselI}\left (-\frac {k^4+4 s k^3+6 s^2 k^2+4 s^3 k+s^4-\sqrt {\left (b^2+2 k b+k^2-4 a d\right ) (k+s)^4 \log ^2\left (e^{k+s}\right )}}{(k+s)^4},2 \sqrt {-\frac {a c \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^4+4 s k^3+6 s^2 k^2+4 s^3 k+s^4+\sqrt {\left (b^2+2 k b+k^2-4 a d\right ) (k+s)^4 \log ^2\left (e^{k+s}\right )}}{(k+s)^4},2 \sqrt {-\frac {a c \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 ) c_1\right ) \sqrt {-\frac {a c \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 )}{2 a \left (K_{\frac {\sqrt {\left (b^2+2 k b+k^2-4 a d\right ) (k+s)^4 \log ^2\left (e^{k+s}\right )}}{(k+s)^4}}\left (2 \sqrt {-\frac {a c \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 )-(-1)^{\frac {k^4+4 s k^3+6 s^2 k^2+4 s^3 k+s^4+\sqrt {\left (b^2+2 k b+k^2-4 a d\right ) (k+s)^4 \log ^2\left (e^{k+s}\right )}}{(k+s)^4}} \operatorname {BesselI}\left (\frac {\sqrt {\left (b^2+2 k b+k^2-4 a d\right ) (k+s)^4 \log ^2\left (e^{k+s}\right )}}{(k+s)^4},2 \sqrt {-\frac {a c \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\right )} y(x)\to \frac {e^{-k x} \left (-(b+k) (k+s)^3 \sqrt {-\frac {a c \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}} \operatorname {BesselI}\left (\frac {\sqrt {\left (b^2+2 k b+k^2-4 a d\right ) (k+s)^4 \log ^2\left (e^{k+s}\right )}}{(k+s)^4},2 \sqrt {-\frac {a c \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 )+a c \log ^2\left (e^{k+s}\right ) \left (\left (e^{k+s}\right )^x\right )^{\frac {k+s}{\log \left (e^{k+s}\right )}} \operatorname {BesselI}\left (\frac {k^4+4 s k^3+6 s^2 k^2+4 s^3 k+s^4+\sqrt {\left (b^2+2 k b+k^2-4 a d\right ) (k+s)^4 \log ^2\left (e^{k+s}\right )}}{(k+s)^4},2 \sqrt {-\frac {a c \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 )+a c \log ^2\left (e^{k+s}\right ) \left (\left (e^{k+s}\right )^x\right )^{\frac {k+s}{\log \left (e^{k+s}\right )}} \operatorname {BesselI}\left (\frac {k^4+4 s k^3+6 s^2 k^2+4 s^3 k+s^4+\sqrt {\left (b^2+2 k b+k^2-4 a d\right ) (k+s)^4 \log ^2\left (e^{k+s}\right )}}{(k+s)^4}-2,2 \sqrt {-\frac {a c \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 )}{2 a (k+s)^3 \sqrt {-\frac {a c \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}} \operatorname {BesselI}\left (\frac {\sqrt {\left (b^2+2 k b+k^2-4 a d\right ) (k+s)^4 \log ^2\left (e^{k+s}\right )}}{(k+s)^4},2 \sqrt {-\frac {a c \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 )} \end{align*}