Internal problem ID [10537]
Internal file name [OUTPUT/9485_Monday_June_06_2022_02_50_37_PM_8560361/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 1.2.6-4. Equations with
cotangent.
Problem number: 40.
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]
\[ \boxed {y^{\prime }-y^{2}+2 a b \cot \left (x a \right ) y=-a^{2}+b^{2}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= y^{2}-2 a b \cot \left (x a \right ) y +b^{2}-a^{2} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = y^{2}-2 a b \cot \left (x a \right ) y +b^{2}-a^{2} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-a^{2}+b^{2}\), \(f_1(x)=-2 a b \cot \left (x a \right )\) 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 &=-2 a b \cot \left (x a \right )\\ f_2^2 f_0 &=-a^{2}+b^{2} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} u^{\prime \prime }\left (x \right )+2 a b \cot \left (x a \right ) u^{\prime }\left (x \right )+\left (-a^{2}+b^{2}\right ) u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \sin \left (x a \right )^{-b +\frac {1}{2}} \left (\operatorname {LegendreP}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right ) c_{1} +\operatorname {LegendreQ}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right ) c_{2} \right ) \] The above shows that \[ u^{\prime }\left (x \right ) = -\frac {2 \sin \left (x a \right )^{-b +\frac {1}{2}} \left (\left (-\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}\, \cos \left (x a \right ) \sin \left (x a \right )+a \left (-\frac {\cos \left (x a \right ) \sin \left (x a \right )}{2}+\left (\cos \left (x a \right )-1\right ) \left (\cos \left (x a \right )+1\right ) \cot \left (x a \right ) \left (b -\frac {1}{2}\right )\right )\right ) c_{1} \operatorname {LegendreP}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right )+\left (-\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}\, \cos \left (x a \right ) \sin \left (x a \right )+a \left (-\frac {\cos \left (x a \right ) \sin \left (x a \right )}{2}+\left (\cos \left (x a \right )-1\right ) \left (\cos \left (x a \right )+1\right ) \cot \left (x a \right ) \left (b -\frac {1}{2}\right )\right )\right ) c_{2} \operatorname {LegendreQ}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right )-\left (-\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}+\left (b -1\right ) a \right ) \left (\operatorname {LegendreP}\left (\frac {a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right ) c_{1} +\operatorname {LegendreQ}\left (\frac {a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right ) c_{2} \right ) \sin \left (x a \right )\right )}{2 \cos \left (x a \right )^{2}-2} \] Using the above in (1) gives the solution \[ y = \frac {2 \left (-\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}\, \cos \left (x a \right ) \sin \left (x a \right )+a \left (-\frac {\cos \left (x a \right ) \sin \left (x a \right )}{2}+\left (\cos \left (x a \right )-1\right ) \left (\cos \left (x a \right )+1\right ) \cot \left (x a \right ) \left (b -\frac {1}{2}\right )\right )\right ) c_{1} \operatorname {LegendreP}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right )+2 \left (-\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}\, \cos \left (x a \right ) \sin \left (x a \right )+a \left (-\frac {\cos \left (x a \right ) \sin \left (x a \right )}{2}+\left (\cos \left (x a \right )-1\right ) \left (\cos \left (x a \right )+1\right ) \cot \left (x a \right ) \left (b -\frac {1}{2}\right )\right )\right ) c_{2} \operatorname {LegendreQ}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right )-2 \left (-\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}+\left (b -1\right ) a \right ) \left (\operatorname {LegendreP}\left (\frac {a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right ) c_{1} +\operatorname {LegendreQ}\left (\frac {a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right ) c_{2} \right ) \sin \left (x a \right )}{\left (2 \cos \left (x a \right )^{2}-2\right ) \left (\operatorname {LegendreP}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right ) c_{1} +\operatorname {LegendreQ}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right ) c_{2} \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {\csc \left (x a \right ) \left (c_{3} \cos \left (x a \right ) \left (a b +\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}\right ) \operatorname {LegendreP}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right )+\cos \left (x a \right ) \left (a b +\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}\right ) \operatorname {LegendreQ}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right )+\left (\operatorname {LegendreP}\left (\frac {a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right ) c_{3} +\operatorname {LegendreQ}\left (\frac {a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right )\right ) \left (-\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}+\left (b -1\right ) a \right )\right )}{\operatorname {LegendreP}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right ) c_{3} +\operatorname {LegendreQ}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\csc \left (x a \right ) \left (c_{3} \cos \left (x a \right ) \left (a b +\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}\right ) \operatorname {LegendreP}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right )+\cos \left (x a \right ) \left (a b +\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}\right ) \operatorname {LegendreQ}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right )+\left (\operatorname {LegendreP}\left (\frac {a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right ) c_{3} +\operatorname {LegendreQ}\left (\frac {a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right )\right ) \left (-\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}+\left (b -1\right ) a \right )\right )}{\operatorname {LegendreP}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right ) c_{3} +\operatorname {LegendreQ}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {\csc \left (x a \right ) \left (c_{3} \cos \left (x a \right ) \left (a b +\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}\right ) \operatorname {LegendreP}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right )+\cos \left (x a \right ) \left (a b +\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}\right ) \operatorname {LegendreQ}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right )+\left (\operatorname {LegendreP}\left (\frac {a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right ) c_{3} +\operatorname {LegendreQ}\left (\frac {a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right )\right ) \left (-\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}+\left (b -1\right ) a \right )\right )}{\operatorname {LegendreP}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right ) c_{3} +\operatorname {LegendreQ}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (x a \right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2}+2 a b \cot \left (x a \right ) y=-a^{2}+b^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{2}-2 a b \cot \left (x a \right ) y+b^{2}-a^{2} \end {array} \]
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) = -2*a*b*cot(a*x)*(diff(y(x), x))+(a^2-b^2)*y(x), y(x)` *** Subleve 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 <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach <- heuristic approach successful <- hypergeometric successful <- special function solution successful Change of variables used: [x = 1/a*arcsin(t)] Linear ODE actually solved: (-a^2*t+b^2*t)*u(t)+(-2*a^2*b*t^2-a^2*t^2+2*a^2*b)*diff(u(t),t)+(-a^2*t^3+a^2*t)*diff(diff(u(t),t),t) = 0 <- change of variables successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 291
dsolve(diff(y(x),x)=y(x)^2-2*a*b*cot(a*x)*y(x)+b^2-a^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\cos \left (a x \right ) \left (a b +\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}\right ) \operatorname {LegendreP}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )+c_{1} \cos \left (a x \right ) \left (a b +\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}\right ) \operatorname {LegendreQ}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )+\left (\operatorname {LegendreQ}\left (\frac {a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right ) c_{1} +\operatorname {LegendreP}\left (\frac {a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )\right ) \left (-\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}+\left (b -1\right ) a \right )\right ) \csc \left (a x \right )}{\operatorname {LegendreQ}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right ) c_{1} +\operatorname {LegendreP}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==y[x]^2-2*a*b*Cot[a*x]*y[x]+b^2-a^2,y[x],x,IncludeSingularSolutions -> True]
Not solved