61.3.15 problem 15
Internal
problem
ID
[12099]
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
Date
solved
:
Tuesday, January 28, 2025 at 07:01:52 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=a \,{\mathrm e}^{\mu x} y^{2}+a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x} y-b \lambda \,{\mathrm e}^{\lambda x} \end{align*}
✓ Solution by Maple
Time used: 0.003 (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 = \frac {-6 \left (-\frac {2 \left (\mu +\frac {\lambda }{2}\right ) \left (\lambda +\mu \right ) {\mathrm e}^{\frac {{\mathrm e}^{x \left (\lambda +\mu \right )} a b -4 x \left (\lambda +\mu \right ) \left (\mu +\frac {3 \lambda }{4}\right )}{2 \lambda +2 \mu }}}{3}+a \,{\mathrm e}^{\frac {{\mathrm e}^{x \left (\lambda +\mu \right )} a b -2 x \left (\mu +\frac {\lambda }{2}\right ) \left (\lambda +\mu \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 )-c_{1} \left (\left (-\lambda ^{2}-3 \mu \lambda -2 \mu ^{2}\right ) {\mathrm e}^{\frac {{\mathrm e}^{x \left (\lambda +\mu \right )} a b -4 x \left (\lambda +\mu \right ) \left (\mu +\frac {3 \lambda }{4}\right )}{2 \lambda +2 \mu }}+a b \left (\left (\lambda +2 \mu \right ) {\mathrm e}^{\frac {{\mathrm e}^{x \left (\lambda +\mu \right )} a b -2 x \left (\mu +\frac {\lambda }{2}\right ) \left (\lambda +\mu \right )}{2 \lambda +2 \mu }}+a \,{\mathrm e}^{\frac {{\mathrm e}^{x \left (\lambda +\mu \right )} a b +x \lambda \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} b \right )\right ) \left (\lambda +\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 )-12 c_{1} \left (\frac {2 \lambda }{3}+\mu \right ) \left (\mu +\frac {\lambda }{2}\right )^{2} {\mathrm e}^{-\frac {\left (3 \lambda +4 \mu \right ) x}{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 {{\mathrm e}^{x \left (\lambda +\mu \right )} a b +x \lambda \left (\lambda +\mu \right )}{\lambda +\mu }} a}{\left (4 \,{\mathrm e}^{\frac {{\mathrm e}^{x \left (\lambda +\mu \right )} a b -2 x \left (\frac {3 \lambda }{2}+\mu \right ) \left (\lambda +\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 )+c_{1} \left (\lambda +\mu \right ) \left (\left (\lambda +2 \mu \right ) {\mathrm e}^{\frac {{\mathrm e}^{x \left (\lambda +\mu \right )} a b -2 x \left (\frac {3 \lambda }{2}+\mu \right ) \left (\lambda +\mu \right )}{2 \lambda +2 \mu }}+b a \,{\mathrm e}^{\frac {{\mathrm e}^{x \left (\lambda +\mu \right )} a b -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: 5.191 (sec). Leaf size: 902
DSolve[D[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],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*}