Internal problem ID [10555]
Internal file name [OUTPUT/9503_Monday_June_06_2022_02_59_53_PM_14938955/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: 58.
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]
\[ \boxed {y^{\prime }-y^{2}=-\frac {\lambda ^{2}}{2}-\frac {3 \lambda ^{2} \tan \left (\lambda x \right )^{2}}{4}+a \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{n}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= y^{2}-\frac {\lambda ^{2}}{2}-\frac {3 \lambda ^{2} \tan \left (\lambda x \right )^{2}}{4}+a \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{n} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = y^{2}-\frac {\lambda ^{2}}{2}-\frac {3 \lambda ^{2} \tan \left (\lambda x \right )^{2}}{4}+a \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{n} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-\frac {\lambda ^{2}}{2}-\frac {3 \lambda ^{2} \tan \left (\lambda x \right )^{2}}{4}+a \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{n}\), \(f_1(x)=0\) 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 &=-\frac {\lambda ^{2}}{2}-\frac {3 \lambda ^{2} \tan \left (\lambda x \right )^{2}}{4}+a \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{n} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} u^{\prime \prime }\left (x \right )+\left (-\frac {\lambda ^{2}}{2}-\frac {3 \lambda ^{2} \tan \left (\lambda x \right )^{2}}{4}+a \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{n}\right ) u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \frac {-\csc \left (\frac {\pi \left (n +3\right )}{n +2}\right ) c_{1} \operatorname {BesselI}\left (-\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \pi \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {1}{2 n +4}}+c_{2} \sin \left (\lambda x \right ) \operatorname {BesselI}\left (\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \Gamma \left (\frac {n +3}{n +2}\right )^{2} \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{-\frac {1}{2 n +4}} \left (n +2\right )}{\sqrt {\cos \left (\lambda x \right )}\, \left (n +2\right ) \Gamma \left (\frac {n +3}{n +2}\right )} \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\left (\Gamma \left (\frac {n +3}{n +2}\right )^{2} \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {n +1}{2 n +4}} c_{2} \cos \left (\lambda x \right )^{2} \left (n +2\right )^{2} \operatorname {BesselI}\left (\frac {n +3}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right )+\frac {c_{2} \operatorname {BesselI}\left (\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \Gamma \left (\frac {n +3}{n +2}\right )^{2} \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{-\frac {1}{2 n +4}} \left (\cos \left (\lambda x \right )^{2}+1\right ) \left (n +2\right )}{2}-\pi \left (\frac {\sin \left (\lambda x \right ) \operatorname {BesselI}\left (-\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {1}{2 n +4}}}{2}+\left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {n +3}{2 n +4}} \cos \left (\lambda x \right ) \operatorname {BesselI}\left (\frac {n +1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \cot \left (\lambda x \right ) \left (n +2\right )\right ) \csc \left (\frac {\pi \left (n +3\right )}{n +2}\right ) c_{1} \right ) \lambda }{\cos \left (\lambda x \right )^{\frac {3}{2}} \left (n +2\right ) \Gamma \left (\frac {n +3}{n +2}\right )} \] Using the above in (1) gives the solution \[ y = -\frac {\left (\Gamma \left (\frac {n +3}{n +2}\right )^{2} \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {n +1}{2 n +4}} c_{2} \cos \left (\lambda x \right )^{2} \left (n +2\right )^{2} \operatorname {BesselI}\left (\frac {n +3}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right )+\frac {c_{2} \operatorname {BesselI}\left (\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \Gamma \left (\frac {n +3}{n +2}\right )^{2} \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{-\frac {1}{2 n +4}} \left (\cos \left (\lambda x \right )^{2}+1\right ) \left (n +2\right )}{2}-\pi \left (\frac {\sin \left (\lambda x \right ) \operatorname {BesselI}\left (-\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {1}{2 n +4}}}{2}+\left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {n +3}{2 n +4}} \cos \left (\lambda x \right ) \operatorname {BesselI}\left (\frac {n +1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \cot \left (\lambda x \right ) \left (n +2\right )\right ) \csc \left (\frac {\pi \left (n +3\right )}{n +2}\right ) c_{1} \right ) \lambda }{\cos \left (\lambda x \right ) \left (-\csc \left (\frac {\pi \left (n +3\right )}{n +2}\right ) c_{1} \operatorname {BesselI}\left (-\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \pi \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {1}{2 n +4}}+c_{2} \sin \left (\lambda x \right ) \operatorname {BesselI}\left (\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \Gamma \left (\frac {n +3}{n +2}\right )^{2} \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{-\frac {1}{2 n +4}} \left (n +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 {\lambda \left (-2 \Gamma \left (\frac {n +3}{n +2}\right )^{2} \sin \left (\frac {\pi \left (n +3\right )}{n +2}\right ) \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {n +1}{2 n +4}} \cos \left (\lambda x \right ) \left (n +2\right )^{2} \operatorname {BesselI}\left (\frac {n +3}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right )+2 \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {n +3}{2 n +4}} \pi c_{3} \cot \left (\lambda x \right ) \left (n +2\right ) \operatorname {BesselI}\left (\frac {n +1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right )+\tan \left (\lambda x \right ) c_{3} \operatorname {BesselI}\left (-\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \pi \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {1}{2 n +4}}-\Gamma \left (\frac {n +3}{n +2}\right )^{2} \sin \left (\frac {\pi \left (n +3\right )}{n +2}\right ) \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{-\frac {1}{2 n +4}} \operatorname {BesselI}\left (\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \left (\cos \left (\lambda x \right )+\sec \left (\lambda x \right )\right ) \left (n +2\right )\right )}{-2 c_{3} \operatorname {BesselI}\left (-\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \pi \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {1}{2 n +4}}+2 \operatorname {BesselI}\left (\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \Gamma \left (\frac {n +3}{n +2}\right )^{2} \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{-\frac {1}{2 n +4}} \sin \left (\lambda x \right ) \sin \left (\frac {\pi \left (n +3\right )}{n +2}\right ) \left (n +2\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\lambda \left (-2 \Gamma \left (\frac {n +3}{n +2}\right )^{2} \sin \left (\frac {\pi \left (n +3\right )}{n +2}\right ) \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {n +1}{2 n +4}} \cos \left (\lambda x \right ) \left (n +2\right )^{2} \operatorname {BesselI}\left (\frac {n +3}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right )+2 \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {n +3}{2 n +4}} \pi c_{3} \cot \left (\lambda x \right ) \left (n +2\right ) \operatorname {BesselI}\left (\frac {n +1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right )+\tan \left (\lambda x \right ) c_{3} \operatorname {BesselI}\left (-\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \pi \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {1}{2 n +4}}-\Gamma \left (\frac {n +3}{n +2}\right )^{2} \sin \left (\frac {\pi \left (n +3\right )}{n +2}\right ) \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{-\frac {1}{2 n +4}} \operatorname {BesselI}\left (\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \left (\cos \left (\lambda x \right )+\sec \left (\lambda x \right )\right ) \left (n +2\right )\right )}{-2 c_{3} \operatorname {BesselI}\left (-\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \pi \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {1}{2 n +4}}+2 \operatorname {BesselI}\left (\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \Gamma \left (\frac {n +3}{n +2}\right )^{2} \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{-\frac {1}{2 n +4}} \sin \left (\lambda x \right ) \sin \left (\frac {\pi \left (n +3\right )}{n +2}\right ) \left (n +2\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {\lambda \left (-2 \Gamma \left (\frac {n +3}{n +2}\right )^{2} \sin \left (\frac {\pi \left (n +3\right )}{n +2}\right ) \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {n +1}{2 n +4}} \cos \left (\lambda x \right ) \left (n +2\right )^{2} \operatorname {BesselI}\left (\frac {n +3}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right )+2 \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {n +3}{2 n +4}} \pi c_{3} \cot \left (\lambda x \right ) \left (n +2\right ) \operatorname {BesselI}\left (\frac {n +1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right )+\tan \left (\lambda x \right ) c_{3} \operatorname {BesselI}\left (-\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \pi \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {1}{2 n +4}}-\Gamma \left (\frac {n +3}{n +2}\right )^{2} \sin \left (\frac {\pi \left (n +3\right )}{n +2}\right ) \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{-\frac {1}{2 n +4}} \operatorname {BesselI}\left (\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \left (\cos \left (\lambda x \right )+\sec \left (\lambda x \right )\right ) \left (n +2\right )\right )}{-2 c_{3} \operatorname {BesselI}\left (-\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \pi \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{\frac {1}{2 n +4}}+2 \operatorname {BesselI}\left (\frac {1}{n +2}, 2 \sqrt {-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}}\right ) \Gamma \left (\frac {n +3}{n +2}\right )^{2} \left (-\frac {a \sin \left (\lambda x \right )^{n +2}}{\lambda ^{2} \left (n +2\right )^{2}}\right )^{-\frac {1}{2 n +4}} \sin \left (\lambda x \right ) \sin \left (\frac {\pi \left (n +3\right )}{n +2}\right ) \left (n +2\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2}=-\frac {\lambda ^{2}}{2}-\frac {3 \lambda ^{2} \tan \left (\lambda x \right )^{2}}{4}+a \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{n} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{2}-\frac {\lambda ^{2}}{2}-\frac {3 \lambda ^{2} \tan \left (\lambda x \right )^{2}}{4}+a \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{n} \end {array} \]
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 Special trying Riccati sub-methods: trying Riccati_symmetries trying Riccati to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = ((1/2)*lambda^2+(3/4)*lambda^2*tan(lambda*x)^2-a*cos(lambda*x)^2*sin(l 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 symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying an equivalence, under non-integer power transformations, to LODEs admitting Liouvillian solutions. -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Whittaker -> hyper3: Equivalence to 1F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 0F1 ODE <- Whittaker successful <- special function solution successful Change of variables used: [x = arccos(t)/lambda] Linear ODE actually solved: (4*a*(-t^2+1)^(1/2*n)*t^4+lambda^2*t^2-3*lambda^2)*u(t)-4*t^3*lambda^2*diff(u(t),t)+(-4*lambda^2*t^4+4*lambda^2*t^2)*dif <- change of variables successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 1194
dsolve(diff(y(x),x)=y(x)^2-1/2*lambda^2-3/4*lambda^2*tan(lambda*x)^2+a*cos(lambda*x)^2*sin(lambda*x)^n,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==y[x]^2-1/2*\[Lambda]^2-3/4*\[Lambda]^2*Tan[\[Lambda]*x]^2+a*Cos[\[Lambda]*x]^2*Sin[\[Lambda]*x]^n,y[x],x,IncludeSingularSolutions -> True]
Not solved