Internal problem ID [10517]
Internal file name [OUTPUT/9465_Monday_June_06_2022_02_44_12_PM_8959918/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: 20.
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]
\[ \boxed {2 y^{\prime }-\left (\lambda +a -a \cos \left (\lambda x \right )\right ) y^{2}=\lambda -a -a \cos \left (\lambda x \right )} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= -\frac {a \,y^{2} \cos \left (\lambda x \right )}{2}+\frac {a \,y^{2}}{2}+\frac {\lambda \,y^{2}}{2}+\frac {\lambda }{2}-\frac {a}{2}-\frac {a \cos \left (\lambda x \right )}{2} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -\frac {a \,y^{2} \cos \left (\lambda x \right )}{2}+\frac {a \,y^{2}}{2}+\frac {\lambda \,y^{2}}{2}+\frac {\lambda }{2}-\frac {a}{2}-\frac {a \cos \left (\lambda x \right )}{2} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {\lambda }{2}-\frac {a}{2}-\frac {a \cos \left (\lambda x \right )}{2}\), \(f_1(x)=0\) and \(f_2(x)=-\frac {a \cos \left (\lambda x \right )}{2}+\frac {a}{2}+\frac {\lambda }{2}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\left (-\frac {a \cos \left (\lambda x \right )}{2}+\frac {a}{2}+\frac {\lambda }{2}\right ) 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' &=\frac {a \sin \left (\lambda x \right ) \lambda }{2}\\ f_1 f_2 &=0\\ f_2^2 f_0 &=\left (-\frac {a \cos \left (\lambda x \right )}{2}+\frac {a}{2}+\frac {\lambda }{2}\right )^{2} \left (\frac {\lambda }{2}-\frac {a}{2}-\frac {a \cos \left (\lambda x \right )}{2}\right ) \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \left (-\frac {a \cos \left (\lambda x \right )}{2}+\frac {a}{2}+\frac {\lambda }{2}\right ) u^{\prime \prime }\left (x \right )-\frac {a \sin \left (\lambda x \right ) \lambda u^{\prime }\left (x \right )}{2}+\left (-\frac {a \cos \left (\lambda x \right )}{2}+\frac {a}{2}+\frac {\lambda }{2}\right )^{2} \left (\frac {\lambda }{2}-\frac {a}{2}-\frac {a \cos \left (\lambda x \right )}{2}\right ) u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \frac {\sin \left (\frac {\lambda x}{2}\right ) {\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }} \left (i c_{2} \lambda \left (\int {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \left (2 a +\csc \left (\frac {\lambda x}{2}\right )^{2} \lambda \right )d x \right )+2 c_{1} \right )}{2} \] The above shows that \[ u^{\prime }\left (x \right ) = -\frac {\csc \left (\frac {\lambda x}{2}\right ) \left (a \cos \left (\lambda x \right )-a -\lambda \right ) \left (i \sin \left (\lambda x \right ) \left (\int {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \left (2 a +\csc \left (\frac {\lambda x}{2}\right )^{2} \lambda \right )d x \right ) c_{2} \lambda \,{\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }}+4 i {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{2 \lambda }} c_{2} \lambda +2 \sin \left (\lambda x \right ) c_{1} {\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }}\right )}{8} \] Using the above in (1) gives the solution \[ y = \frac {\csc \left (\frac {\lambda x}{2}\right ) \left (a \cos \left (\lambda x \right )-a -\lambda \right ) \left (i \sin \left (\lambda x \right ) \left (\int {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \left (2 a +\csc \left (\frac {\lambda x}{2}\right )^{2} \lambda \right )d x \right ) c_{2} \lambda \,{\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }}+4 i {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{2 \lambda }} c_{2} \lambda +2 \sin \left (\lambda x \right ) c_{1} {\mathrm e}^{-\frac {a \cos \left (\lambda x \right )}{2 \lambda }}\right ) {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{2 \lambda }}}{4 \left (-\frac {a \cos \left (\lambda x \right )}{2}+\frac {a}{2}+\frac {\lambda }{2}\right ) \sin \left (\frac {\lambda x}{2}\right ) \left (i c_{2} \lambda \left (\int {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \left (2 a +\csc \left (\frac {\lambda x}{2}\right )^{2} \lambda \right )d x \right )+2 c_{1} \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {2 \left (i \sin \left (\lambda x \right ) c_{3} -\frac {\lambda \left (\int {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \left (2 a +\csc \left (\frac {\lambda x}{2}\right )^{2} \lambda \right )d x \right ) \sin \left (\lambda x \right )}{2}-2 \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda \right ) \csc \left (\frac {\lambda x}{2}\right )^{2}}{4 i c_{3} -2 \lambda \left (\int {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \left (2 a +\csc \left (\frac {\lambda x}{2}\right )^{2} \lambda \right )d x \right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {2 \left (i \sin \left (\lambda x \right ) c_{3} -\frac {\lambda \left (\int {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \left (2 a +\csc \left (\frac {\lambda x}{2}\right )^{2} \lambda \right )d x \right ) \sin \left (\lambda x \right )}{2}-2 \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda \right ) \csc \left (\frac {\lambda x}{2}\right )^{2}}{4 i c_{3} -2 \lambda \left (\int {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \left (2 a +\csc \left (\frac {\lambda x}{2}\right )^{2} \lambda \right )d x \right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {2 \left (i \sin \left (\lambda x \right ) c_{3} -\frac {\lambda \left (\int {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \left (2 a +\csc \left (\frac {\lambda x}{2}\right )^{2} \lambda \right )d x \right ) \sin \left (\lambda x \right )}{2}-2 \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \lambda \right ) \csc \left (\frac {\lambda x}{2}\right )^{2}}{4 i c_{3} -2 \lambda \left (\int {\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \left (2 a +\csc \left (\frac {\lambda x}{2}\right )^{2} \lambda \right )d x \right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 y^{\prime }-\left (\lambda +a -a \cos \left (\lambda x \right )\right ) y^{2}=\lambda -a -a \cos \left (\lambda x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\left (\lambda +a -a \cos \left (\lambda x \right )\right ) y^{2}}{2}+\frac {\lambda }{2}-\frac {a}{2}-\frac {a \cos \left (\lambda x \right )}{2} \end {array} \]
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 sub-methods: trying Riccati_symmetries trying Riccati to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = -a*lambda*sin(lambda*x)*(diff(y(x), x))/(-lambda-a+a*cos(lambda*x))-(1 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 No special function solution was found. <- Kovacics algorithm successful Change of variables used: [x = arccos(t)/lambda] Linear ODE actually solved: (4*a^3*t^3-4*a^3*t^2-12*a^2*lambda*t^2-4*a^3*t+8*a^2*lambda*t+12*a*lambda^2*t+4*a^3+4*a^2*lambda-4*a*lambda^2-4*lambda^3 <- change of variables successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 122
dsolve(2*diff(y(x),x)=(lambda+a-a*cos(lambda*x))*y(x)^2+lambda-a-a*cos(lambda*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {-\cot \left (\frac {x \lambda }{2}\right ) \lambda \left (\int {\mathrm e}^{\frac {a \cos \left (x \lambda \right )}{\lambda }} \operatorname {csgn}\left (\sin \left (\frac {x \lambda }{2}\right )\right ) \left (\csc \left (\frac {x \lambda }{2}\right )^{2} \lambda +2 a \right )d x \right ) c_{1} -2 \csc \left (\frac {x \lambda }{2}\right )^{2} \operatorname {csgn}\left (\sin \left (\frac {x \lambda }{2}\right )\right ) {\mathrm e}^{\frac {a \cos \left (x \lambda \right )}{\lambda }} c_{1} \lambda +2 i \cot \left (\frac {x \lambda }{2}\right )}{\lambda \left (\int {\mathrm e}^{\frac {a \cos \left (x \lambda \right )}{\lambda }} \operatorname {csgn}\left (\sin \left (\frac {x \lambda }{2}\right )\right ) \left (\csc \left (\frac {x \lambda }{2}\right )^{2} \lambda +2 a \right )d x \right ) c_{1} -2 i} \]
✓ Solution by Mathematica
Time used: 34.139 (sec). Leaf size: 234
DSolve[2*y'[x]==(\[Lambda]+a-a*Cos[\[Lambda]*x])*y[x]^2+\[Lambda]-a-a*Cos[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {2 \left (c_1 \cot \left (\frac {\lambda x}{2}\right ) \int _1^xe^{-\frac {2 a \sin ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} \left (\lambda \csc ^2\left (\frac {1}{2} \lambda K[1]\right )+2 a\right )dK[1]+2 c_1 \csc ^2\left (\frac {\lambda x}{2}\right ) e^{-\frac {2 a \sin ^2\left (\frac {\lambda x}{2}\right )}{\lambda }}+\cot \left (\frac {\lambda x}{2}\right )\right )}{2+2 c_1 \int _1^xe^{-\frac {2 a \sin ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} \left (\lambda \csc ^2\left (\frac {1}{2} \lambda K[1]\right )+2 a\right )dK[1]} \\ y(x)\to \frac {1}{2} \csc ^2\left (\frac {\lambda x}{2}\right ) \left (-\frac {4 e^{-\frac {2 a \sin ^2\left (\frac {\lambda x}{2}\right )}{\lambda }}}{\int _1^xe^{-\frac {2 a \sin ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} \left (\lambda \csc ^2\left (\frac {1}{2} \lambda K[1]\right )+2 a\right )dK[1]}-\sin (\lambda x)\right ) \\ \end{align*}