problem 15

Internal problem ID [10422]
Internal ﬁle name [OUTPUT/9370_Monday_June_06_2022_02_19_07_PM_49470304/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: 15.
ODE order: 1.
ODE degree: 1.

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

3.15.1 Solving as riccati ode

In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= a \,{\mathrm e}^{\mu x} y^{2}+a b \,{\mathrm e}^{x \left (\lambda +\mu \right )} y -b \lambda \,{\mathrm e}^{\lambda x} \end {align*}

This is a Riccati ODE. Comparing the ODE to solve \[ y' = a \,{\mathrm e}^{\mu x} y^{2}+a b \,{\mathrm e}^{\lambda x} {\mathrm e}^{\mu x} y -b \lambda \,{\mathrm e}^{\lambda x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-b \lambda \,{\mathrm e}^{\lambda x}\), \(f_1(x)=a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}\) and \(f_2(x)=a \,{\mathrm e}^{\mu x}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{a \,{\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' &=a \mu \,{\mathrm e}^{\mu x}\\ f_1 f_2 &=a^{2} b \,{\mathrm e}^{x \left (\lambda +\mu \right )} {\mathrm e}^{\mu x}\\ f_2^2 f_0 &=-{\mathrm e}^{\lambda x} {\mathrm e}^{2 \mu x} a^{2} b \lambda \end {align*}

Substituting the above terms back in equation (2) gives \begin {align*} a \,{\mathrm e}^{\mu x} u^{\prime \prime }\left (x \right )-\left (a \mu \,{\mathrm e}^{\mu x}+a^{2} b \,{\mathrm e}^{x \left (\lambda +\mu \right )} {\mathrm e}^{\mu x}\right ) u^{\prime }\left (x \right )-{\mathrm e}^{\lambda x} {\mathrm e}^{2 \mu x} a^{2} b \lambda u \left (x \right ) &=0 \end {align*}

\[ u \left (x \right ) = 4 \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\frac {3 \lambda }{2}+\mu \right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} \left (\mu +\frac {\lambda }{2}\right )^{2} c_{2} \operatorname {WhittakerM}\left (\frac {\lambda +2 \mu }{2 \lambda +2 \mu }, \frac {2 \lambda +3 \mu }{2 \lambda +2 \mu }, \frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )+c_{2} \left (\lambda +\mu \right ) \left (\left (\lambda +2 \mu \right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\frac {3 \lambda }{2}+\mu \right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }}+b a \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-x \lambda \left (\lambda +\mu \right )}{2 \lambda +2 \mu }}\right ) \operatorname {WhittakerM}\left (-\frac {\lambda }{2 \lambda +2 \mu }, \frac {2 \lambda +3 \mu }{2 \lambda +2 \mu }, \frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )+c_{1} {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }} \] The above shows that \[ u^{\prime }\left (x \right ) = 6 \left (\mu +\frac {\lambda }{2}\right ) \left (-\frac {2 \left (\mu +\frac {\lambda }{2}\right ) \left (\lambda +\mu \right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\frac {3 \lambda }{2}+\mu \right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }}}{3}+a \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-x \lambda \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} b \left (\frac {2 \lambda }{3}+\mu \right )\right ) c_{2} \operatorname {WhittakerM}\left (\frac {\lambda +2 \mu }{2 \lambda +2 \mu }, \frac {2 \lambda +3 \mu }{2 \lambda +2 \mu }, \frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )+\left (\left (-\lambda ^{2}-3 \mu \lambda -2 \mu ^{2}\right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\frac {3 \lambda }{2}+\mu \right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }}+\left (a \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}+2 \left (\mu +\frac {\lambda }{2}\right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} b +2 \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-x \lambda \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} \left (\mu +\frac {\lambda }{2}\right )\right ) b a \right ) c_{2} \left (\lambda +\mu \right ) \operatorname {WhittakerM}\left (-\frac {\lambda }{2 \lambda +2 \mu }, \frac {2 \lambda +3 \mu }{2 \lambda +2 \mu }, \frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )+12 \left (\mu +\frac {\lambda }{2}\right )^{2} \left (\frac {2 \lambda }{3}+\mu \right ) c_{2} {\mathrm e}^{-\frac {x \left (3 \lambda +2 \mu \right )}{2}} \left (\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }\right )^{\frac {3 \lambda +4 \mu }{2 \lambda +2 \mu }}+a b c_{1} {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}+x \left (\lambda +\mu \right )^{2}}{\lambda +\mu }} \] Using the above in (1) gives the solution \[ y = -\frac {\left (6 \left (\mu +\frac {\lambda }{2}\right ) \left (-\frac {2 \left (\mu +\frac {\lambda }{2}\right ) \left (\lambda +\mu \right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\frac {3 \lambda }{2}+\mu \right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }}}{3}+a \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-x \lambda \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} b \left (\frac {2 \lambda }{3}+\mu \right )\right ) c_{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 (\left (-\lambda ^{2}-3 \mu \lambda -2 \mu ^{2}\right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\frac {3 \lambda }{2}+\mu \right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }}+\left (a \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}+2 \left (\mu +\frac {\lambda }{2}\right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} b +2 \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-x \lambda \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} \left (\mu +\frac {\lambda }{2}\right )\right ) b a \right ) c_{2} \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 \left (\mu +\frac {\lambda }{2}\right )^{2} \left (\frac {2 \lambda }{3}+\mu \right ) c_{2} {\mathrm e}^{-\frac {x \left (3 \lambda +2 \mu \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 }}+a b c_{1} {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}+x \left (\lambda +\mu \right )^{2}}{\lambda +\mu }}\right ) {\mathrm e}^{-\mu x}}{a \left (4 \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\frac {3 \lambda }{2}+\mu \right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} \left (\mu +\frac {\lambda }{2}\right )^{2} c_{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_{2} \left (\lambda +\mu \right ) \left (\left (\lambda +2 \mu \right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\frac {3 \lambda }{2}+\mu \right ) x \left (\lambda +\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 )+c_{1} {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }}\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 (6 \left (\mu +\frac {\lambda }{2}\right ) \left (-\frac {2 \left (\mu +\frac {\lambda }{2}\right ) \left (\lambda +\mu \right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\frac {3 \lambda }{2}+\mu \right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }}}{3}+a \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-x \lambda \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} b \left (\frac {2 \lambda }{3}+\mu \right )\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 )}-2 \left (\frac {3 \lambda }{2}+\mu \right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }}+\left (a \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}+2 \left (\mu +\frac {\lambda }{2}\right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} b +2 \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-x \lambda \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} \left (\mu +\frac {\lambda }{2}\right )\right ) b a \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 \left (\mu +\frac {\lambda }{2}\right )^{2} \left (\frac {2 \lambda }{3}+\mu \right ) {\mathrm e}^{-\frac {x \left (3 \lambda +2 \mu \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 }}+a b c_{3} {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}+x \left (\lambda +\mu \right )^{2}}{\lambda +\mu }}\right ) {\mathrm e}^{-\mu x}}{\left (4 \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\frac {3 \lambda }{2}+\mu \right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} \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 ) \left (\left (\lambda +2 \mu \right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\frac {3 \lambda }{2}+\mu \right ) x \left (\lambda +\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 )+c_{3} {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }}\right ) a} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\left (6 \left (\mu +\frac {\lambda }{2}\right ) \left (-\frac {2 \left (\mu +\frac {\lambda }{2}\right ) \left (\lambda +\mu \right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\frac {3 \lambda }{2}+\mu \right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }}}{3}+a \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-x \lambda \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} b \left (\frac {2 \lambda }{3}+\mu \right )\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 )}-2 \left (\frac {3 \lambda }{2}+\mu \right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }}+\left (a \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}+2 \left (\mu +\frac {\lambda }{2}\right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} b +2 \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-x \lambda \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} \left (\mu +\frac {\lambda }{2}\right )\right ) b a \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 \left (\mu +\frac {\lambda }{2}\right )^{2} \left (\frac {2 \lambda }{3}+\mu \right ) {\mathrm e}^{-\frac {x \left (3 \lambda +2 \mu \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 }}+a b c_{3} {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}+x \left (\lambda +\mu \right )^{2}}{\lambda +\mu }}\right ) {\mathrm e}^{-\mu x}}{\left (4 \,{\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\frac {3 \lambda }{2}+\mu \right ) x \left (\lambda +\mu \right )}{2 \lambda +2 \mu }} \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 ) \left (\left (\lambda +2 \mu \right ) {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-2 \left (\frac {3 \lambda }{2}+\mu \right ) x \left (\lambda +\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 )+c_{3} {\mathrm e}^{\frac {a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}}{\lambda +\mu }}\right ) a} \\ \end{align*}

Veriﬁcation of solutions

3.15.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-a \,{\mathrm e}^{\mu x} y^{2}-a b \,{\mathrm e}^{x \left (\lambda +\mu \right )} y=-b \lambda \,{\mathrm e}^{\lambda x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=a \,{\mathrm e}^{\mu x} y^{2}+a b \,{\mathrm e}^{x \left (\lambda +\mu \right )} y-b \lambda \,{\mathrm e}^{\lambda x} \end {array} \]

Maple trace Kovacic algorithm successful

`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) = (b*a*exp(lambda*x+mu*x)+mu)*(diff(y(x), x))+b*exp(x*mu)*exp(lambda*x)* 
      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 
         trying a quadrature 
         checking if the LODE has constant coefficients 
         checking if the LODE is of Euler type 
         trying a symmetry of the form [xi=0, eta=F(x)] 
         checking if the LODE is missing y 
         -> Trying a Liouvillian solution using Kovacics algorithm 
            A Liouvillian solution exists 
            Reducible group (found an exponential solution) 
            Group is reducible, not completely reducible 
         <- Kovacics algorithm successful 
         Change of variables used: 
            [x = ln(t)/(lambda+mu)] 
         Linear ODE actually solved: 
            -a*b*lambda*u(t)+(-a*b*lambda*t-a*b*mu*t+lambda^2+lambda*mu)*diff(u(t),t)+(lambda^2*t+2*lambda*mu*t+mu^2*t)*diff(diff(u( 
      <- change of variables successful 
   <- Riccati to 2nd Order successful`

\[ 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} \]

\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*}

3.15 problem 15

3.15.1 Solving as riccati ode

3.15.2 Maple step by step solution