Internal problem ID [10474]
Internal file name [OUTPUT/9422_Monday_June_06_2022_02_31_42_PM_84409171/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: 27.
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]
\[ \boxed {y^{\prime }-y^{2}=-2 a b +a \lambda +b \lambda -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}-b \left (b +\lambda \right ) \coth \left (\lambda x \right )^{2}} \]
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 -b^{2} \coth \left (\lambda x \right )^{2}-b \coth \left (\lambda x \right )^{2} \lambda -2 a b +a \lambda +b \lambda +y^{2} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -a^{2} \tanh \left (\lambda x \right )^{2}-a \tanh \left (\lambda x \right )^{2} \lambda -b^{2} \coth \left (\lambda x \right )^{2}-b \coth \left (\lambda x \right )^{2} \lambda -2 a b +a \lambda +b \lambda +y^{2} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-a^{2} \tanh \left (\lambda x \right )^{2}-a \tanh \left (\lambda x \right )^{2} \lambda -b^{2} \coth \left (\lambda x \right )^{2}-b \coth \left (\lambda x \right )^{2} \lambda -2 a b +a \lambda +b \lambda \), \(f_1(x)=0\) 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 -b^{2} \coth \left (\lambda x \right )^{2}-b \coth \left (\lambda x \right )^{2} \lambda -2 a b +a \lambda +b \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 -b^{2} \coth \left (\lambda x \right )^{2}-b \coth \left (\lambda x \right )^{2} \lambda -2 a b +a \lambda +b \lambda \right ) u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \operatorname {csch}\left (\lambda x \right )^{\frac {-a -b}{\lambda }} \left (c_{2} \coth \left (\lambda x \right )^{\frac {a +\lambda }{\lambda }} \operatorname {hypergeom}\left (\left [1, \frac {1}{2}-\frac {b}{\lambda }\right ], \left [\frac {3}{2}+\frac {a}{\lambda }\right ], \coth \left (\lambda x \right )^{2}\right )+c_{1} \coth \left (\lambda x \right )^{-\frac {a}{\lambda }} \left (-\operatorname {csch}\left (\lambda x \right )^{2}\right )^{\frac {a +b}{\lambda }}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {4 \operatorname {csch}\left (\lambda x \right )^{\frac {-a -b}{\lambda }} \left (\frac {\coth \left (\lambda x \right )^{\frac {a +\lambda }{\lambda }} \left (\frac {3 \lambda }{2}+a \right ) c_{2} \left (\left (b -\lambda \right ) \coth \left (\lambda x \right )+\tanh \left (\lambda x \right ) \left (a +\lambda \right )\right ) \operatorname {hypergeom}\left (\left [1, \frac {1}{2}-\frac {b}{\lambda }\right ], \left [\frac {3}{2}+\frac {a}{\lambda }\right ], \coth \left (\lambda x \right )^{2}\right )}{2}+\coth \left (\lambda x \right )^{\frac {a +\lambda }{\lambda }} \left (\coth \left (\lambda x \right )+1\right ) \left (\coth \left (\lambda x \right )-1\right ) c_{2} \left (-\frac {\lambda }{2}+b \right ) \coth \left (\lambda x \right ) \lambda \operatorname {hypergeom}\left (\left [2, \frac {3}{2}-\frac {b}{\lambda }\right ], \left [\frac {a}{\lambda }+\frac {5}{2}\right ], \coth \left (\lambda x \right )^{2}\right )+\left (\left (-a -b \right ) \coth \left (\lambda x \right )^{\frac {\lambda -a}{\lambda }}+\coth \left (\lambda x \right )^{-\frac {a}{\lambda }} \left (\left (a +\frac {b}{2}\right ) \coth \left (\lambda x \right )-\frac {a \tanh \left (\lambda x \right )}{2}\right )\right ) \left (\frac {3 \lambda }{2}+a \right ) c_{1} \left (-\operatorname {csch}\left (\lambda x \right )^{2}\right )^{\frac {a +b}{\lambda }}\right )}{2 a +3 \lambda } \] Using the above in (1) gives the solution \[ y = -\frac {4 \left (\frac {\coth \left (\lambda x \right )^{\frac {a +\lambda }{\lambda }} \left (\frac {3 \lambda }{2}+a \right ) c_{2} \left (\left (b -\lambda \right ) \coth \left (\lambda x \right )+\tanh \left (\lambda x \right ) \left (a +\lambda \right )\right ) \operatorname {hypergeom}\left (\left [1, \frac {1}{2}-\frac {b}{\lambda }\right ], \left [\frac {3}{2}+\frac {a}{\lambda }\right ], \coth \left (\lambda x \right )^{2}\right )}{2}+\coth \left (\lambda x \right )^{\frac {a +\lambda }{\lambda }} \left (\coth \left (\lambda x \right )+1\right ) \left (\coth \left (\lambda x \right )-1\right ) c_{2} \left (-\frac {\lambda }{2}+b \right ) \coth \left (\lambda x \right ) \lambda \operatorname {hypergeom}\left (\left [2, \frac {3}{2}-\frac {b}{\lambda }\right ], \left [\frac {a}{\lambda }+\frac {5}{2}\right ], \coth \left (\lambda x \right )^{2}\right )+\left (\left (-a -b \right ) \coth \left (\lambda x \right )^{\frac {\lambda -a}{\lambda }}+\coth \left (\lambda x \right )^{-\frac {a}{\lambda }} \left (\left (a +\frac {b}{2}\right ) \coth \left (\lambda x \right )-\frac {a \tanh \left (\lambda x \right )}{2}\right )\right ) \left (\frac {3 \lambda }{2}+a \right ) c_{1} \left (-\operatorname {csch}\left (\lambda x \right )^{2}\right )^{\frac {a +b}{\lambda }}\right )}{\left (2 a +3 \lambda \right ) \left (c_{2} \coth \left (\lambda x \right )^{\frac {a +\lambda }{\lambda }} \operatorname {hypergeom}\left (\left [1, \frac {1}{2}-\frac {b}{\lambda }\right ], \left [\frac {3}{2}+\frac {a}{\lambda }\right ], \coth \left (\lambda x \right )^{2}\right )+c_{1} \coth \left (\lambda x \right )^{-\frac {a}{\lambda }} \left (-\operatorname {csch}\left (\lambda x \right )^{2}\right )^{\frac {a +b}{\lambda }}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {-4 \left (-\frac {\lambda }{2}+b \right ) \coth \left (\lambda x \right )^{\frac {2 a +2 \lambda }{\lambda }} \lambda \operatorname {csch}\left (\lambda x \right )^{2} \operatorname {hypergeom}\left (\left [2, -\frac {2 b -3 \lambda }{2 \lambda }\right ], \left [\frac {2 a +5 \lambda }{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )+\left (\left (\left (-3 a -3 b \right ) \lambda -2 a b \right ) \coth \left (\lambda x \right )^{\frac {2 a +2 \lambda }{\lambda }}-2 \coth \left (\lambda x \right )^{\frac {2 a +\lambda }{\lambda }} \left (-\frac {5 \,\operatorname {sech}\left (\lambda x \right ) \left (a +\frac {3 \lambda }{5}\right ) \lambda \,\operatorname {csch}\left (\lambda x \right )}{2}+a^{2} \tanh \left (\lambda x \right )\right )\right ) \operatorname {hypergeom}\left (\left [1, \frac {-2 b +\lambda }{2 \lambda }\right ], \left [\frac {2 a +3 \lambda }{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )+2 \left (b \coth \left (\lambda x \right )+a \tanh \left (\lambda x \right )\right ) \left (\frac {3 \lambda }{2}+a \right ) c_{3} \left (-\operatorname {csch}\left (\lambda x \right )^{2}\right )^{\frac {a +b}{\lambda }}}{\left (\operatorname {hypergeom}\left (\left [1, \frac {-2 b +\lambda }{2 \lambda }\right ], \left [\frac {2 a +3 \lambda }{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right ) \coth \left (\lambda x \right )^{\frac {2 a +\lambda }{\lambda }}+c_{3} \left (-\operatorname {csch}\left (\lambda x \right )^{2}\right )^{\frac {a +b}{\lambda }}\right ) \left (2 a +3 \lambda \right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {-4 \left (-\frac {\lambda }{2}+b \right ) \coth \left (\lambda x \right )^{\frac {2 a +2 \lambda }{\lambda }} \lambda \operatorname {csch}\left (\lambda x \right )^{2} \operatorname {hypergeom}\left (\left [2, -\frac {2 b -3 \lambda }{2 \lambda }\right ], \left [\frac {2 a +5 \lambda }{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )+\left (\left (\left (-3 a -3 b \right ) \lambda -2 a b \right ) \coth \left (\lambda x \right )^{\frac {2 a +2 \lambda }{\lambda }}-2 \coth \left (\lambda x \right )^{\frac {2 a +\lambda }{\lambda }} \left (-\frac {5 \,\operatorname {sech}\left (\lambda x \right ) \left (a +\frac {3 \lambda }{5}\right ) \lambda \,\operatorname {csch}\left (\lambda x \right )}{2}+a^{2} \tanh \left (\lambda x \right )\right )\right ) \operatorname {hypergeom}\left (\left [1, \frac {-2 b +\lambda }{2 \lambda }\right ], \left [\frac {2 a +3 \lambda }{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )+2 \left (b \coth \left (\lambda x \right )+a \tanh \left (\lambda x \right )\right ) \left (\frac {3 \lambda }{2}+a \right ) c_{3} \left (-\operatorname {csch}\left (\lambda x \right )^{2}\right )^{\frac {a +b}{\lambda }}}{\left (\operatorname {hypergeom}\left (\left [1, \frac {-2 b +\lambda }{2 \lambda }\right ], \left [\frac {2 a +3 \lambda }{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right ) \coth \left (\lambda x \right )^{\frac {2 a +\lambda }{\lambda }}+c_{3} \left (-\operatorname {csch}\left (\lambda x \right )^{2}\right )^{\frac {a +b}{\lambda }}\right ) \left (2 a +3 \lambda \right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {-4 \left (-\frac {\lambda }{2}+b \right ) \coth \left (\lambda x \right )^{\frac {2 a +2 \lambda }{\lambda }} \lambda \operatorname {csch}\left (\lambda x \right )^{2} \operatorname {hypergeom}\left (\left [2, -\frac {2 b -3 \lambda }{2 \lambda }\right ], \left [\frac {2 a +5 \lambda }{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )+\left (\left (\left (-3 a -3 b \right ) \lambda -2 a b \right ) \coth \left (\lambda x \right )^{\frac {2 a +2 \lambda }{\lambda }}-2 \coth \left (\lambda x \right )^{\frac {2 a +\lambda }{\lambda }} \left (-\frac {5 \,\operatorname {sech}\left (\lambda x \right ) \left (a +\frac {3 \lambda }{5}\right ) \lambda \,\operatorname {csch}\left (\lambda x \right )}{2}+a^{2} \tanh \left (\lambda x \right )\right )\right ) \operatorname {hypergeom}\left (\left [1, \frac {-2 b +\lambda }{2 \lambda }\right ], \left [\frac {2 a +3 \lambda }{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right )+2 \left (b \coth \left (\lambda x \right )+a \tanh \left (\lambda x \right )\right ) \left (\frac {3 \lambda }{2}+a \right ) c_{3} \left (-\operatorname {csch}\left (\lambda x \right )^{2}\right )^{\frac {a +b}{\lambda }}}{\left (\operatorname {hypergeom}\left (\left [1, \frac {-2 b +\lambda }{2 \lambda }\right ], \left [\frac {2 a +3 \lambda }{2 \lambda }\right ], \coth \left (\lambda x \right )^{2}\right ) \coth \left (\lambda x \right )^{\frac {2 a +\lambda }{\lambda }}+c_{3} \left (-\operatorname {csch}\left (\lambda x \right )^{2}\right )^{\frac {a +b}{\lambda }}\right ) \left (2 a +3 \lambda \right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2}=-2 a b +a \lambda +b \lambda -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}-b \left (b +\lambda \right ) \coth \left (\lambda x \right )^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{2}+a \lambda +b \lambda -2 a b -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}-b \left (b +\lambda \right ) \coth \left (\lambda x \right )^{2} \end {array} \]
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) = (b^2*coth(lambda*x)^2+b*coth(lambda*x)^2*lambda+a^2*tanh(lambda*x)^2+a 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 -> Whittaker -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach <- heuristic approach successful <- hypergeometric 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 = arccoth(t)/lambda] Linear ODE actually solved: (-b^2*t^4-b*lambda*t^4-2*a*b*t^2+a*lambda*t^2+b*lambda*t^2-a^2-a*lambda)*u(t)+(2*lambda^2*t^5-2*lambda^2*t^3)*diff(u(t), <- change of variables successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 289
dsolve(diff(y(x),x)=y(x)^2+a*lambda+b*lambda-2*a*b-a*(a+lambda)*tanh(lambda*x)^2-b*(b+lambda)*coth(lambda*x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-4 c_{1} \lambda \left (b -\frac {\lambda }{2}\right ) \coth \left (x \lambda \right )^{\frac {2 a +2 \lambda }{\lambda }} \operatorname {csch}\left (x \lambda \right )^{2} \operatorname {hypergeom}\left (\left [2, -\frac {2 b -3 \lambda }{2 \lambda }\right ], \left [\frac {2 a +5 \lambda }{2 \lambda }\right ], \coth \left (x \lambda \right )^{2}\right )-2 c_{1} \left (\left (\left (\frac {3 a}{2}+\frac {3 b}{2}\right ) \lambda +a b \right ) \coth \left (x \lambda \right )^{\frac {2 a +2 \lambda }{\lambda }}+\left (-\frac {5 \,\operatorname {sech}\left (x \lambda \right ) \lambda \left (a +\frac {3 \lambda }{5}\right ) \operatorname {csch}\left (x \lambda \right )}{2}+a^{2} \tanh \left (x \lambda \right )\right ) \coth \left (x \lambda \right )^{\frac {2 a +\lambda }{\lambda }}\right ) \operatorname {hypergeom}\left (\left [1, \frac {-2 b +\lambda }{2 \lambda }\right ], \left [\frac {2 a +3 \lambda }{2 \lambda }\right ], \coth \left (x \lambda \right )^{2}\right )+2 \left (a \tanh \left (x \lambda \right )+\coth \left (x \lambda \right ) b \right ) \left (a +\frac {3 \lambda }{2}\right ) \left (-\operatorname {csch}\left (x \lambda \right )^{2}\right )^{\frac {a +b}{\lambda }}}{\left (\operatorname {hypergeom}\left (\left [1, \frac {-2 b +\lambda }{2 \lambda }\right ], \left [\frac {2 a +3 \lambda }{2 \lambda }\right ], \coth \left (x \lambda \right )^{2}\right ) c_{1} \coth \left (x \lambda \right )^{\frac {2 a +\lambda }{\lambda }}+\left (-\operatorname {csch}\left (x \lambda \right )^{2}\right )^{\frac {a +b}{\lambda }}\right ) \left (2 a +3 \lambda \right )} \]
✓ Solution by Mathematica
Time used: 40.238 (sec). Leaf size: 493
DSolve[y'[x]==y[x]^2+a*\[Lambda]+b*\[Lambda]-2*a*b-a*(a+\[Lambda])*Tanh[\[Lambda]*x]^2-b*(b+\[Lambda])*Coth[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {(a+b) \left (e^{2 \lambda x}\right )^{\frac {a+b}{\lambda }} \left (\frac {2 \lambda \left (a \left (e^{2 \lambda x}-1\right )^2+b \left (e^{2 \lambda x}+1\right )^2\right ) \left (e^{2 \lambda x}\right )^{-\frac {a+b}{\lambda }} \operatorname {AppellF1}\left (-\frac {a+b}{\lambda },-\frac {2 b}{\lambda },-\frac {2 a}{\lambda },-\frac {a+b-\lambda }{\lambda },e^{2 x \lambda },-e^{2 x \lambda }\right )}{(a+b) \left (e^{2 \lambda x}-1\right ) \left (e^{2 \lambda x}+1\right )}+4 \lambda \left (e^{2 \lambda x}\right )^{-\frac {a+b}{\lambda }} \operatorname {AppellF1}\left (-\frac {a+b}{\lambda },-\frac {2 b}{\lambda },-\frac {2 a}{\lambda },-\frac {a+b-\lambda }{\lambda },e^{2 x \lambda },-e^{2 x \lambda }\right )+\frac {8 \lambda \left (e^{2 \lambda x}\right )^{-\frac {a+b-\lambda }{\lambda }} \left (a \operatorname {AppellF1}\left (1-\frac {a+b}{\lambda },-\frac {2 b}{\lambda },1-\frac {2 a}{\lambda },-\frac {a+b-2 \lambda }{\lambda },e^{2 x \lambda },-e^{2 x \lambda }\right )-b \operatorname {AppellF1}\left (1-\frac {a+b}{\lambda },1-\frac {2 b}{\lambda },-\frac {2 a}{\lambda },-\frac {a+b-2 \lambda }{\lambda },e^{2 x \lambda },-e^{2 x \lambda }\right )\right )}{-a-b+\lambda }-\frac {2 c_1 \left (a \left (e^{2 \lambda x}-1\right )^2+b \left (e^{2 \lambda x}+1\right )^2\right )}{e^{4 \lambda x}-1}\right )}{2 \left (-\lambda \operatorname {AppellF1}\left (-\frac {a+b}{\lambda },-\frac {2 b}{\lambda },-\frac {2 a}{\lambda },-\frac {a+b-\lambda }{\lambda },e^{2 x \lambda },-e^{2 x \lambda }\right )+c_1 (a+b) \left (e^{2 \lambda x}\right )^{\frac {a+b}{\lambda }}\right )} \\ y(x)\to \frac {a \left (e^{2 \lambda x}-1\right )^2+b \left (e^{2 \lambda x}+1\right )^2}{e^{4 \lambda x}-1} \\ \end{align*}