Internal problem ID [10554]
Internal file name [OUTPUT/9502_Monday_June_06_2022_02_59_50_PM_45559588/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 1.2.6-5. Equations containing
combinations of trigonometric functions.
Problem number: 57.
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]
\[ \boxed {y^{\prime }-y^{2}=2 b a +a \lambda +\lambda b +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2}+b \left (\lambda -b \right ) \cot \left (\lambda x \right )^{2}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= -a^{2} \tan \left (\lambda x \right )^{2}+a \tan \left (\lambda x \right )^{2} \lambda -b^{2} \cot \left (\lambda x \right )^{2}+b \cot \left (\lambda x \right )^{2} \lambda +2 b a +a \lambda +\lambda b +y^{2} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -a^{2} \tan \left (\lambda x \right )^{2}+a \tan \left (\lambda x \right )^{2} \lambda -b^{2} \cot \left (\lambda x \right )^{2}+b \cot \left (\lambda x \right )^{2} \lambda +2 b a +a \lambda +\lambda b +y^{2} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-a^{2} \tan \left (\lambda x \right )^{2}+a \tan \left (\lambda x \right )^{2} \lambda -b^{2} \cot \left (\lambda x \right )^{2}+b \cot \left (\lambda x \right )^{2} \lambda +2 b a +a \lambda +\lambda b\), \(f_1(x)=0\) 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} \tan \left (\lambda x \right )^{2}+a \tan \left (\lambda x \right )^{2} \lambda -b^{2} \cot \left (\lambda x \right )^{2}+b \cot \left (\lambda x \right )^{2} \lambda +2 b a +a \lambda +\lambda b \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} u^{\prime \prime }\left (x \right )+\left (-a^{2} \tan \left (\lambda x \right )^{2}+a \tan \left (\lambda x \right )^{2} \lambda -b^{2} \cot \left (\lambda x \right )^{2}+b \cot \left (\lambda x \right )^{2} \lambda +2 b a +a \lambda +\lambda b \right ) u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = c_{1} \cos \left (\lambda x \right )^{\frac {a}{\lambda }} \sin \left (\lambda x \right )^{\frac {b}{\lambda }}+c_{2} \cos \left (\lambda x \right )^{\frac {\lambda -a}{\lambda }} \sin \left (\lambda x \right )^{\frac {\lambda -b}{\lambda }} \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {-4 \cos \left (\lambda x \right )^{\frac {2 \lambda -a}{\lambda }} \sin \left (\lambda x \right )^{\frac {2 \lambda -b}{\lambda }} c_{2} \lambda \left (a +b -\lambda \right ) \operatorname {hypergeom}\left (\left [2, \frac {2 \lambda -a -b}{\lambda }\right ], \left [-\frac {-5 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )-2 \left (-\frac {3 \lambda }{2}+a \right ) \left (-\cos \left (\lambda x \right )^{\frac {\lambda -a}{\lambda }} \left (\left (\lambda -b \right ) \cot \left (\lambda x \right )+\tan \left (\lambda x \right ) \left (a -\lambda \right )\right ) \sin \left (\lambda x \right )^{\frac {\lambda -b}{\lambda }} c_{2} \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )+\cos \left (\lambda x \right )^{\frac {a}{\lambda }} \sin \left (\lambda x \right )^{\frac {b}{\lambda }} c_{1} \left (\tan \left (\lambda x \right ) a -b \cot \left (\lambda x \right )\right )\right )}{-3 \lambda +2 a} \] Using the above in (1) gives the solution \[ y = -\frac {-4 \cos \left (\lambda x \right )^{\frac {2 \lambda -a}{\lambda }} \sin \left (\lambda x \right )^{\frac {2 \lambda -b}{\lambda }} c_{2} \lambda \left (a +b -\lambda \right ) \operatorname {hypergeom}\left (\left [2, \frac {2 \lambda -a -b}{\lambda }\right ], \left [-\frac {-5 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )-2 \left (-\frac {3 \lambda }{2}+a \right ) \left (-\cos \left (\lambda x \right )^{\frac {\lambda -a}{\lambda }} \left (\left (\lambda -b \right ) \cot \left (\lambda x \right )+\tan \left (\lambda x \right ) \left (a -\lambda \right )\right ) \sin \left (\lambda x \right )^{\frac {\lambda -b}{\lambda }} c_{2} \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )+\cos \left (\lambda x \right )^{\frac {a}{\lambda }} \sin \left (\lambda x \right )^{\frac {b}{\lambda }} c_{1} \left (\tan \left (\lambda x \right ) a -b \cot \left (\lambda x \right )\right )\right )}{\left (-3 \lambda +2 a \right ) \left (c_{1} \cos \left (\lambda x \right )^{\frac {a}{\lambda }} \sin \left (\lambda x \right )^{\frac {b}{\lambda }}+c_{2} \cos \left (\lambda x \right )^{\frac {\lambda -a}{\lambda }} \sin \left (\lambda x \right )^{\frac {\lambda -b}{\lambda }} \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {4 \lambda \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{2} \left (a +b -\lambda \right ) \operatorname {hypergeom}\left (\left [2, \frac {2 \lambda -a -b}{\lambda }\right ], \left [-\frac {-5 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )+\left (\left (6 \lambda ^{2}+\left (-7 a -3 b \right ) \lambda +2 b a \right ) \cos \left (\lambda x \right )^{2}-2 \sin \left (\lambda x \right )^{2} a^{2}+5 a \lambda -3 \lambda ^{2}\right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )+2 \sin \left (\lambda x \right )^{\frac {2 b}{\lambda }} c_{3} \left (\tan \left (\lambda x \right ) a -b \cot \left (\lambda x \right )\right ) \left (-\frac {3 \lambda }{2}+a \right ) \cos \left (\lambda x \right )^{\frac {2 a}{\lambda }}}{\left (-3 \lambda +2 a \right ) \left (c_{3} \cos \left (\lambda x \right )^{\frac {2 a}{\lambda }} \sin \left (\lambda x \right )^{\frac {2 b}{\lambda }}+\cos \left (\lambda x \right ) \sin \left (\lambda x \right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {4 \lambda \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{2} \left (a +b -\lambda \right ) \operatorname {hypergeom}\left (\left [2, \frac {2 \lambda -a -b}{\lambda }\right ], \left [-\frac {-5 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )+\left (\left (6 \lambda ^{2}+\left (-7 a -3 b \right ) \lambda +2 b a \right ) \cos \left (\lambda x \right )^{2}-2 \sin \left (\lambda x \right )^{2} a^{2}+5 a \lambda -3 \lambda ^{2}\right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )+2 \sin \left (\lambda x \right )^{\frac {2 b}{\lambda }} c_{3} \left (\tan \left (\lambda x \right ) a -b \cot \left (\lambda x \right )\right ) \left (-\frac {3 \lambda }{2}+a \right ) \cos \left (\lambda x \right )^{\frac {2 a}{\lambda }}}{\left (-3 \lambda +2 a \right ) \left (c_{3} \cos \left (\lambda x \right )^{\frac {2 a}{\lambda }} \sin \left (\lambda x \right )^{\frac {2 b}{\lambda }}+\cos \left (\lambda x \right ) \sin \left (\lambda x \right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {4 \lambda \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{2} \left (a +b -\lambda \right ) \operatorname {hypergeom}\left (\left [2, \frac {2 \lambda -a -b}{\lambda }\right ], \left [-\frac {-5 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )+\left (\left (6 \lambda ^{2}+\left (-7 a -3 b \right ) \lambda +2 b a \right ) \cos \left (\lambda x \right )^{2}-2 \sin \left (\lambda x \right )^{2} a^{2}+5 a \lambda -3 \lambda ^{2}\right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )+2 \sin \left (\lambda x \right )^{\frac {2 b}{\lambda }} c_{3} \left (\tan \left (\lambda x \right ) a -b \cot \left (\lambda x \right )\right ) \left (-\frac {3 \lambda }{2}+a \right ) \cos \left (\lambda x \right )^{\frac {2 a}{\lambda }}}{\left (-3 \lambda +2 a \right ) \left (c_{3} \cos \left (\lambda x \right )^{\frac {2 a}{\lambda }} \sin \left (\lambda x \right )^{\frac {2 b}{\lambda }}+\cos \left (\lambda x \right ) \sin \left (\lambda x \right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {-3 \lambda +2 a}{2 \lambda }\right ], \cos \left (\lambda x \right )^{2}\right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2}=2 b a +a \lambda +\lambda b +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2}+b \left (\lambda -b \right ) \cot \left (\lambda x \right )^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{2}+a \lambda +\lambda b +2 b a +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2}+b \left (\lambda -b \right ) \cot \left (\lambda x \right )^{2} \end {array} \]
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*tan(lambda*x)^2-a*tan(lambda*x)^2*lambda+b^2*cot(lambda*x)^2-b*co 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 for at least one hypergeometric solution is achieved - returning with no uncomputed integrals <- Kovacics algorithm successful Change of variables used: [x = 1/lambda*arccos(t)] Linear ODE actually solved: (a^2*t^4+2*a*b*t^4+b^2*t^4-2*a^2*t^2-2*a*b*t^2+a*lambda*t^2-b*lambda*t^2+a^2-a*lambda)*u(t)+(-lambda^2*t^5+lambda^2*t^3) <- change of variables successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 268
dsolve(diff(y(x),x)=y(x)^2+lambda*a+lambda*b+2*a*b+a*(lambda-a)*tan(lambda*x)^2+b*(lambda-b)*cot(lambda*x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {4 c_{1} \lambda \cos \left (x \lambda \right )^{2} \sin \left (x \lambda \right )^{2} \left (b -\lambda +a \right ) \operatorname {hypergeom}\left (\left [2, \frac {2 \lambda -b -a}{\lambda }\right ], \left [-\frac {2 a -5 \lambda }{2 \lambda }\right ], \cos \left (x \lambda \right )^{2}\right )-2 c_{1} \left (\left (-3 \lambda ^{2}+\left (\frac {7 a}{2}+\frac {3 b}{2}\right ) \lambda -a b \right ) \cos \left (x \lambda \right )^{2}+a^{2} \sin \left (x \lambda \right )^{2}-\frac {5 \left (a -\frac {3 \lambda }{5}\right ) \lambda }{2}\right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {2 a -3 \lambda }{2 \lambda }\right ], \cos \left (x \lambda \right )^{2}\right )+2 \left (a -\frac {3 \lambda }{2}\right ) \sin \left (x \lambda \right )^{\frac {2 b}{\lambda }} \left (a \tan \left (x \lambda \right )-b \cot \left (x \lambda \right )\right ) \cos \left (x \lambda \right )^{\frac {2 a}{\lambda }}}{\left (2 a -3 \lambda \right ) \left (c_{1} \cos \left (x \lambda \right ) \sin \left (x \lambda \right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {2 a -3 \lambda }{2 \lambda }\right ], \cos \left (x \lambda \right )^{2}\right )+\cos \left (x \lambda \right )^{\frac {2 a}{\lambda }} \sin \left (x \lambda \right )^{\frac {2 b}{\lambda }}\right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==y[x]^2+\[Lambda]*a+\[Lambda]*b+2*a*b+a*(\[Lambda]-a)*Tan[\[Lambda]*x]^2+b*(\[Lambda]-b)*Cot[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
Not solved