problem 18

Internal problem ID [10465]
Internal ﬁle name [OUTPUT/9413_Monday_June_06_2022_02_28_29_PM_65894832/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 1.2.4-2. Equations with hyperbolic tangent and cotangent.
Problem number: 18.
ODE order: 1.
ODE degree: 1.

\[ \boxed {y^{\prime }-y^{2}=a \lambda -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}} \]

6.1.1 Solving as riccati ode

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

This is a Riccati ODE. Comparing the ODE to solve \[ y' = -a^{2} \tanh \left (\lambda x \right )^{2}-a \tanh \left (\lambda x \right )^{2} \lambda +a \lambda +y^{2} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-a^{2} \tanh \left (\lambda x \right )^{2}-a \tanh \left (\lambda x \right )^{2} \lambda +a \lambda \), \(f_1(x)=0\) and \(f_2(x)=1\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{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' &=0\\ f_1 f_2 &=0\\ f_2^2 f_0 &=-a^{2} \tanh \left (\lambda x \right )^{2}-a \tanh \left (\lambda x \right )^{2} \lambda +a \lambda \end {align*}

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

\[ u \left (x \right ) = c_{1} \operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )+c_{2} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = -\operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right ) c_{1} \lambda -\operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right ) c_{2} \lambda +\tanh \left (\lambda x \right ) \left (c_{1} \operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )+c_{2} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )\right ) \left (a +\lambda \right ) \] Using the above in (1) gives the solution \[ y = -\frac {-\operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right ) c_{1} \lambda -\operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right ) c_{2} \lambda +\tanh \left (\lambda x \right ) \left (c_{1} \operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )+c_{2} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )\right ) \left (a +\lambda \right )}{c_{1} \operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )+c_{2} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution

\[ y = \frac {\operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right ) c_{3} \lambda +\operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right ) \lambda -\tanh \left (\lambda x \right ) \left (c_{3} \operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )+\operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )\right ) \left (a +\lambda \right )}{c_{3} \operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )+\operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right ) c_{3} \lambda +\operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right ) \lambda -\tanh \left (\lambda x \right ) \left (c_{3} \operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )+\operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )\right ) \left (a +\lambda \right )}{c_{3} \operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )+\operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )} \\ \end{align*}

Veriﬁcation of solutions

6.1.2 Maple step by step solution

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

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 Special 
trying Riccati sub-methods: 
   trying Riccati_symmetries 
   trying Riccati to 2nd Order 
   -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = (a^2*tanh(lambda*x)^2+a*tanh(lambda*x)^2*lambda-a*lambda)*y(x), y(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 
            Solution has integrals. Trying a special function solution free of integrals... 
            -> Trying a solution in terms of special functions: 
               -> Bessel 
               -> elliptic 
               -> Legendre 
               <- Legendre successful 
            <- special function solution successful 
               -> Trying to convert hypergeometric functions to elementary form... 
               <- elementary form could result into a too large expression - returning special function form of solution, free of un 
            <- Kovacics algorithm successful 
         Change of variables used: 
            [x = arctanh(t)/lambda] 
         Linear ODE actually solved: 
            (-a^2*t^2-a*lambda*t^2+a*lambda)*u(t)+(2*lambda^2*t^3-2*lambda^2*t)*diff(u(t),t)+(lambda^2*t^4-2*lambda^2*t^2+lambda^2)* 
      <- change of variables successful 
   <- Riccati to 2nd Order successful`

\[ y \left (x \right ) = \frac {\operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \tanh \left (x \lambda \right )\right ) \lambda +\operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \tanh \left (x \lambda \right )\right ) c_{1} \lambda -\tanh \left (x \lambda \right ) \left (c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (x \lambda \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (x \lambda \right )\right )\right ) \left (a +\lambda \right )}{c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (x \lambda \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (x \lambda \right )\right )} \]

\begin{align*} y(x)\to \frac {a \left (-\lambda \left (e^{2 \lambda x}-1\right ) \operatorname {Hypergeometric2F1}\left (-\frac {2 a}{\lambda },-\frac {a}{\lambda },1-\frac {a}{\lambda },-e^{2 x \lambda }\right )-2 \lambda \left (e^{2 \lambda x}+1\right )^{\frac {2 a}{\lambda }+1}+a c_1 \left (e^{2 \lambda x}-1\right ) \left (e^{2 \lambda x}\right )^{a/\lambda }\right )}{\left (e^{2 \lambda x}+1\right ) \left (-\lambda \operatorname {Hypergeometric2F1}\left (-\frac {2 a}{\lambda },-\frac {a}{\lambda },1-\frac {a}{\lambda },-e^{2 x \lambda }\right )+a c_1 \left (e^{2 \lambda x}\right )^{a/\lambda }\right )} \\ y(x)\to \frac {a \left (e^{2 \lambda x}-1\right )}{e^{2 \lambda x}+1} \\ \end{align*}

6.1 problem 18

6.1.1 Solving as riccati ode

6.1.2 Maple step by step solution