Internal problem ID [10417]
Internal file name [OUTPUT/9365_Monday_June_06_2022_02_19_01_PM_15885292/index.tex]
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: 10.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "riccati"
Maple gives the following as the ode type
[_Riccati]
\[ \boxed {y^{\prime }-b \,{\mathrm e}^{\mu x} y^{2}=a \lambda \,{\mathrm e}^{\lambda x}-a^{2} b \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= b \,{\mathrm e}^{\mu x} y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} b \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = b \,{\mathrm e}^{\mu x} y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} b \,{\mathrm e}^{2 \lambda x} {\mathrm e}^{\mu x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=a \lambda \,{\mathrm e}^{\lambda x}-a^{2} b \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x}\), \(f_1(x)=0\) and \(f_2(x)=b \,{\mathrm e}^{\mu x}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{b \,{\mathrm e}^{\mu x} u} \tag {1} \end {align*}
Using the above substitution in the given ODE results (after some simplification)in a second order ODE to solve for \(u(x)\) which is \begin {align*} f_2 u''(x) -\left ( f_2' + f_1 f_2 \right ) u'(x) + f_2^2 f_0 u(x) &= 0 \tag {2} \end {align*}
But \begin {align*} f_2' &=b \mu \,{\mathrm e}^{\mu x}\\ f_1 f_2 &=0\\ f_2^2 f_0 &=b^{2} {\mathrm e}^{2 \mu x} \left (a \lambda \,{\mathrm e}^{\lambda x}-a^{2} b \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x}\right ) \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} b \,{\mathrm e}^{\mu x} u^{\prime \prime }\left (x \right )-b \mu \,{\mathrm e}^{\mu x} u^{\prime }\left (x \right )+b^{2} {\mathrm e}^{2 \mu x} \left (a \lambda \,{\mathrm e}^{\lambda x}-a^{2} b \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x}\right ) u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{-{\mathrm e}^{2 x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a^{2} b^{2}+{\mathrm e}^{x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a b \lambda -\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{-{\mathrm e}^{2 x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a^{2} b^{2}+{\mathrm e}^{x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a b \lambda -\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{-{\mathrm e}^{2 x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a^{2} b^{2}+{\mathrm e}^{x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a b \lambda -\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) {\mathrm e}^{-\mu x}}{b \operatorname {DESol}\left (\left \{-{\mathrm e}^{2 x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a^{2} b^{2}+{\mathrm e}^{x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a b \lambda -\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{-{\mathrm e}^{2 x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a^{2} b^{2}+{\mathrm e}^{x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a b \lambda -\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) {\mathrm e}^{-\mu x}}{b \operatorname {DESol}\left (\left \{-{\mathrm e}^{2 x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a^{2} b^{2}+{\mathrm e}^{x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a b \lambda -\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{-{\mathrm e}^{2 x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a^{2} b^{2}+{\mathrm e}^{x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a b \lambda -\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) {\mathrm e}^{-\mu x}}{b \operatorname {DESol}\left (\left \{-{\mathrm e}^{2 x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a^{2} b^{2}+{\mathrm e}^{x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a b \lambda -\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{-{\mathrm e}^{2 x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a^{2} b^{2}+{\mathrm e}^{x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a b \lambda -\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) {\mathrm e}^{-\mu x}}{b \operatorname {DESol}\left (\left \{-{\mathrm e}^{2 x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a^{2} b^{2}+{\mathrm e}^{x \left (\lambda +\mu \right )} \textit {\_Y} \left (x \right ) a b \lambda -\mu \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-b \,{\mathrm e}^{\mu x} y^{2}=a \lambda \,{\mathrm e}^{\lambda x}-a^{2} b \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=b \,{\mathrm e}^{\mu x} y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} b \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying Riccati trying Riccati sub-methods: trying Riccati_symmetries trying Riccati to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = mu*(diff(y(x), x))-b*exp(x*mu)*a*(-exp(2*lambda*x+mu*x)*a*b+lambda*exp Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler <- unable to find a useful change of variables trying a symmetry of the form [xi=0, eta=F(x)] trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients trying 2nd order, integrating factor of the form mu(x,y) trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler <- unable to find a useful change of variables trying a symmetry of the form [xi=0, eta=F(x)] trying to convert to an ODE of Bessel type -> Trying a change of variables to reduce to Bernoulli -> Calling odsolve with the ODE`, diff(y(x), x)-(b*exp(x*mu)*y(x)^2+y(x)+x^2*(a*lambda*exp(lambda*x)-a^2*b*exp(2*lambda*x+mu*x))) Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying Riccati trying Riccati sub-methods: trying Riccati_symmetries trying inverse_Riccati trying 1st order ODE linearizable_by_differentiation -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] trying inverse_Riccati trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 4 `, `-> Computing symmetries using: way = 2 `, `-> Computing symmetries using: way = 6`
✗ Solution by Maple
dsolve(diff(y(x),x)=b*exp(mu*x)*y(x)^2+a*lambda*exp(lambda*x)-a^2*b*exp((mu+2*lambda)*x),y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 8.808 (sec). Leaf size: 844
DSolve[y'[x]==b*Exp[\[Mu]*x]*y[x]^2+a*\[Lambda]*Exp[\[Lambda]*x]-a^2*b*Exp[(\[Mu]+2*\[Lambda])*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{\mu (-x)} \left (-2 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 {2 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 {2 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 (\log \left (e^{\lambda +\mu }\right ) \left (2 a b \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}}+\mu \right )+\mu (\lambda +\mu )\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 {2 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 (\log \left (e^{\lambda +\mu }\right ) \left (2 a b \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}}+\mu \right )+\mu (\lambda +\mu )\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 {2 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 b (\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 {2 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 {2 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 {a 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 {2 (\lambda +\mu )^2+\mu \log \left (e^{\lambda +\mu }\right )}{(\lambda +\mu )^2},-\frac {2 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 )^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 {2 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 (\log \left (e^{\lambda +\mu }\right ) \left (2 a b \left (\left (e^{\lambda +\mu }\right )^x\right )^{\frac {\lambda +\mu }{\log \left (e^{\lambda +\mu }\right )}}+\mu \right )+\mu (\lambda +\mu )\right )}{2 b (\lambda +\mu )} \\ \end{align*}