3.15 problem 15

Internal problem ID [10423]

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: 15.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [_Riccati]

\[ \boxed {y^{\prime }-a \,{\mathrm e}^{x \mu } y^{2}-a b \,{\mathrm e}^{x \left (\lambda +\mu \right )} y=-b \lambda \,{\mathrm e}^{\lambda x}} \]

✓ Solution by Maple

Time used: 0.0 (sec). Leaf size: 629

dsolve(diff(y(x),x)=a*exp(mu*x)*y(x)^2+a*b*exp((lambda+mu)*x)*y(x)-b*lambda*exp(lambda*x),y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {-6 \left (-\frac {2 \left (\lambda +\mu \right ) \left (\mu +\frac {\lambda }{2}\right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-4 \left (\lambda +\mu \right ) x \left (\mu +\frac {3 \lambda }{4}\right )}{2 \lambda +2 \mu }}}{3}+a \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\lambda +\mu \right ) x \left (\mu +\frac {\lambda }{2}\right )}{2 \lambda +2 \mu }} b \left (\frac {2 \lambda }{3}+\mu \right )\right ) c_{1} \left (\mu +\frac {\lambda }{2}\right ) \operatorname {WhittakerM}\left (\frac {\lambda +2 \mu }{2 \lambda +2 \mu }, \frac {2 \lambda +3 \mu }{2 \lambda +2 \mu }, \frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )-\left (\left (-\lambda ^{2}-3 \mu \lambda -2 \mu ^{2}\right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-4 \left (\lambda +\mu \right ) x \left (\mu +\frac {3 \lambda }{4}\right )}{2 \lambda +2 \mu }}+\left (\left (\lambda +2 \mu \right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\lambda +\mu \right ) x \left (\mu +\frac {\lambda }{2}\right )}{2 \lambda +2 \mu }}+a \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}+x \lambda \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} b \right ) b a \right ) \left (\lambda +\mu \right ) c_{1} \operatorname {WhittakerM}\left (-\frac {\lambda }{2 \lambda +2 \mu }, \frac {2 \lambda +3 \mu }{2 \lambda +2 \mu }, \frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )-12 \,{\mathrm e}^{-\frac {\left (3 \lambda +4 \mu \right ) x}{2}} \left (\frac {2 \lambda }{3}+\mu \right ) c_{1} \left (\mu +\frac {\lambda }{2}\right )^{2} \left (\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )^{\frac {3 \lambda +4 \mu }{2 \lambda +2 \mu }}-b \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}+x \lambda \left (\lambda +\mu \right )}{\lambda +\mu }} a}{\left (4 \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\lambda +\mu \right ) x \left (\frac {3 \lambda }{2}+\mu \right )}{2 \lambda +2 \mu }} c_{1} \left (\mu +\frac {\lambda }{2}\right )^{2} \operatorname {WhittakerM}\left (\frac {\lambda +2 \mu }{2 \lambda +2 \mu }, \frac {2 \lambda +3 \mu }{2 \lambda +2 \mu }, \frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )+\left (\lambda +\mu \right ) c_{1} \left (\left (\lambda +2 \mu \right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\lambda +\mu \right ) x \left (\frac {3 \lambda }{2}+\mu \right )}{2 \lambda +2 \mu }}+b a \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-x \lambda \left (\lambda +\mu \right )}{2 \lambda +2 \mu }}\right ) \operatorname {WhittakerM}\left (-\frac {\lambda }{2 \lambda +2 \mu }, \frac {2 \lambda +3 \mu }{2 \lambda +2 \mu }, \frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )+{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }}\right ) a} \]

✓ Solution by Mathematica

Time used: 12.587 (sec). Leaf size: 902

DSolve[y'[x]==a*Exp[\[Mu]*x]*y[x]^2+a*b*Exp[(\[Lambda]+\[Mu])*x]*y[x]-b*\[Lambda]*Exp[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {e^{\mu (-x)} \left (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 {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 ) \operatorname {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 {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 )-c_1 (\lambda +\mu ) \left ((\lambda +\mu ) \left (a b \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}}+\mu \right )+\log \left (e^{\lambda +\mu }\right ) \left (\mu -a b \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}}\right )\right ) \operatorname {HypergeometricU}\left (\frac {\lambda +\mu +\log \left (e^{\lambda +\mu }\right )}{2 (\lambda +\mu )},\frac {\mu \log \left (e^{\lambda +\mu }\right )}{(\lambda +\mu )^2}+1,\frac {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 )-(\lambda +\mu ) \left ((\lambda +\mu ) \left (a b \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}}+\mu \right )+\log \left (e^{\lambda +\mu }\right ) \left (\mu -a b \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}}\right )\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 {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 a (\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 {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 \operatorname {HypergeometricU}\left (\frac {\lambda +\mu +\log \left (e^{\lambda +\mu }\right )}{2 (\lambda +\mu )},\frac {\mu \log \left (e^{\lambda +\mu }\right )}{(\lambda +\mu )^2}+1,\frac {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 )} \\ y(x)\to \frac {b 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 )}} \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\log \left (e^{\lambda +\mu }\right )}{\lambda +\mu }+3\right ),\frac {\mu \log \left (e^{\lambda +\mu }\right )}{(\lambda +\mu )^2}+2,\frac {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 )}{2 (\lambda +\mu )^2 \operatorname {HypergeometricU}\left (\frac {\lambda +\mu +\log \left (e^{\lambda +\mu }\right )}{2 (\lambda +\mu )},\frac {\mu \log \left (e^{\lambda +\mu }\right )}{(\lambda +\mu )^2}+1,\frac {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 )}-\frac {e^{\mu (-x)} \left ((\lambda +\mu ) \left (a b \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}}+\mu \right )+\log \left (e^{\lambda +\mu }\right ) \left (\mu -a b \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}}\right )\right )}{2 a (\lambda +\mu )} \\ \end{align*}