Internal problem ID [10512]
Internal file name [OUTPUT/9460_Monday_June_06_2022_02_40_20_PM_50369115/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 1.2.6-2. Equations with
cosine.
Problem number: 15.
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]
\[ \boxed {y^{\prime }-y^{2}=-a^{2}+a \lambda \cos \left (\lambda x \right )+\cos \left (\lambda x \right )^{2} a^{2}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= y^{2}-a^{2}+a \lambda \cos \left (\lambda x \right )+\cos \left (\lambda x \right )^{2} a^{2} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = y^{2}-a^{2}+a \lambda \cos \left (\lambda x \right )+\cos \left (\lambda x \right )^{2} a^{2} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-a^{2}+a \lambda \cos \left (\lambda x \right )+\cos \left (\lambda x \right )^{2} a^{2}\), \(f_1(x)=0\) 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}+a \lambda \cos \left (\lambda x \right )+\cos \left (\lambda x \right )^{2} a^{2} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} u^{\prime \prime }\left (x \right )+\left (-a^{2}+a \lambda \cos \left (\lambda x \right )+\cos \left (\lambda x \right )^{2} a^{2}\right ) u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \left (c_{1} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )+c_{2} \cos \left (\frac {\lambda x}{2}\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = -2 \left (\frac {c_{2} \left (a \cos \left (\lambda x \right )+a +\frac {\lambda }{2}\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{2}+\frac {\left (\cos \left (\lambda x \right )+1\right ) c_{2} \lambda \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{4}+c_{1} \cos \left (\frac {\lambda x}{2}\right ) \left (a \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )+\frac {\lambda \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{2}\right )\right ) {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \sin \left (\frac {\lambda x}{2}\right ) \] Using the above in (1) gives the solution \[ y = \frac {2 \left (\frac {c_{2} \left (a \cos \left (\lambda x \right )+a +\frac {\lambda }{2}\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{2}+\frac {\left (\cos \left (\lambda x \right )+1\right ) c_{2} \lambda \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{4}+c_{1} \cos \left (\frac {\lambda x}{2}\right ) \left (a \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )+\frac {\lambda \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{2}\right )\right ) \sin \left (\frac {\lambda x}{2}\right )}{c_{1} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )+c_{2} \cos \left (\frac {\lambda x}{2}\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {2 \left (\frac {\left (a \cos \left (\lambda x \right )+a +\frac {\lambda }{2}\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{2}+\frac {\left (\cos \left (\lambda x \right )+1\right ) \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \lambda }{4}+c_{3} \cos \left (\frac {\lambda x}{2}\right ) \left (a \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )+\frac {\lambda \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{2}\right )\right ) \sin \left (\frac {\lambda x}{2}\right )}{c_{3} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )+\operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cos \left (\frac {\lambda x}{2}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {2 \left (\frac {\left (a \cos \left (\lambda x \right )+a +\frac {\lambda }{2}\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{2}+\frac {\left (\cos \left (\lambda x \right )+1\right ) \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \lambda }{4}+c_{3} \cos \left (\frac {\lambda x}{2}\right ) \left (a \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )+\frac {\lambda \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{2}\right )\right ) \sin \left (\frac {\lambda x}{2}\right )}{c_{3} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )+\operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cos \left (\frac {\lambda x}{2}\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {2 \left (\frac {\left (a \cos \left (\lambda x \right )+a +\frac {\lambda }{2}\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{2}+\frac {\left (\cos \left (\lambda x \right )+1\right ) \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \lambda }{4}+c_{3} \cos \left (\frac {\lambda x}{2}\right ) \left (a \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )+\frac {\lambda \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )}{2}\right )\right ) \sin \left (\frac {\lambda x}{2}\right )}{c_{3} \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )+\operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cos \left (\frac {\lambda x}{2}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2}=-a^{2}+a \lambda \cos \left (\lambda x \right )+\cos \left (\lambda x \right )^{2} a^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{2}-a^{2}+a \lambda \cos \left (\lambda x \right )+\cos \left (\lambda x \right )^{2} a^{2} \end {array} \]
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-a*lambda*cos(lambda*x)-cos(lambda*x)^2*a^2)*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 -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius <- Heun successful: received ODE is equivalent to the HeunC ODE, case a <> 0, e <> 0, c = 0 <- Kovacics algorithm successful Change of variables used: [x = arccos(t)/lambda] Linear ODE actually solved: (2*a^2*t^2+2*a*lambda*t-2*a^2)*u(t)-2*lambda^2*t*diff(u(t),t)+(-2*lambda^2*t^2+2*lambda^2)*diff(diff(u(t),t),t) = 0 <- change of variables successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 272
dsolve(diff(y(x),x)=y(x)^2-a^2+a*lambda*cos(lambda*x)+a^2*cos(lambda*x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (2 a c_{1} \sin \left (x \lambda \right ) \cos \left (\frac {x \lambda }{2}\right )+c_{1} \lambda \sin \left (\frac {x \lambda }{2}\right )\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (x \lambda \right )}{2}+\frac {1}{2}\right )+2 \sin \left (x \lambda \right ) \left (a \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (x \lambda \right )}{2}+\frac {1}{2}\right )+\frac {\lambda \left (\operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (x \lambda \right )}{2}+\frac {1}{2}\right ) c_{1} \cos \left (\frac {x \lambda }{2}\right )+\operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (x \lambda \right )}{2}+\frac {1}{2}\right )\right )}{2}\right )}{2 \cos \left (\frac {x \lambda }{2}\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (x \lambda \right )}{2}+\frac {1}{2}\right ) c_{1} +2 \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\cos \left (x \lambda \right )}{2}+\frac {1}{2}\right )} \]
✓ Solution by Mathematica
Time used: 3.942 (sec). Leaf size: 131
DSolve[y'[x]==y[x]^2-a^2+a*\[Lambda]*Cos[\[Lambda]*x]+a^2*Cos[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {a c_1 \sin (\lambda x) \int _1^xe^{-\frac {2 a \cos (\lambda K[1])}{\lambda }}dK[1]+a \sin (\lambda x)+c_1 \left (-e^{-\frac {2 a \cos (\lambda x)}{\lambda }}\right )}{1+c_1 \int _1^xe^{-\frac {2 a \cos (\lambda K[1])}{\lambda }}dK[1]} \\ y(x)\to a \sin (\lambda x)-\frac {e^{-\frac {2 a \cos (\lambda x)}{\lambda }}}{\int _1^xe^{-\frac {2 a \cos (\lambda K[1])}{\lambda }}dK[1]} \\ \end{align*}