Internal problem ID [10545]
Internal file name [OUTPUT/9493_Monday_June_06_2022_02_55_37_PM_29498587/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: 48.
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]
\[ \boxed {y^{\prime }-a \sin \left (\lambda x \right ) y^{2}=b \sin \left (\lambda x \right ) \cos \left (\lambda x \right )^{n}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= a \sin \left (\lambda x \right ) y^{2}+b \sin \left (\lambda x \right ) \cos \left (\lambda x \right )^{n} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = a \sin \left (\lambda x \right ) y^{2}+b \sin \left (\lambda x \right ) \cos \left (\lambda x \right )^{n} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=b \sin \left (\lambda x \right ) \cos \left (\lambda x \right )^{n}\), \(f_1(x)=0\) and \(f_2(x)=a \sin \left (\lambda x \right )\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{a \sin \left (\lambda x \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' &=a \cos \left (\lambda x \right ) \lambda \\ f_1 f_2 &=0\\ f_2^2 f_0 &=a^{2} \sin \left (\lambda x \right )^{3} b \cos \left (\lambda x \right )^{n} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} a \sin \left (\lambda x \right ) u^{\prime \prime }\left (x \right )-a \cos \left (\lambda x \right ) \lambda u^{\prime }\left (x \right )+a^{2} \sin \left (\lambda x \right )^{3} b \cos \left (\lambda x \right )^{n} u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \sqrt {\cos \left (\lambda x \right )}\, \left (c_{1} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )+c_{2} \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\lambda \sin \left (\lambda x \right ) \left (\operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right ) \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}} c_{1} +\sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}} \operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right ) c_{2} -c_{1} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )-c_{2} \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )\right )}{\sqrt {\cos \left (\lambda x \right )}} \] Using the above in (1) gives the solution \[ y = -\frac {\lambda \left (\operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right ) \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}} c_{1} +\sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}} \operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right ) c_{2} -c_{1} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )-c_{2} \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )\right )}{\cos \left (\lambda x \right ) a \left (c_{1} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )+c_{2} \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {\left (-\operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right ) \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}} c_{3} -\sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}} \operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )+c_{3} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )+\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )\right ) \lambda \sec \left (\lambda x \right )}{a \left (c_{3} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )+\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (-\operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right ) \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}} c_{3} -\sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}} \operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )+c_{3} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )+\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )\right ) \lambda \sec \left (\lambda x \right )}{a \left (c_{3} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )+\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {\left (-\operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right ) \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}} c_{3} -\sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}} \operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )+c_{3} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )+\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )\right ) \lambda \sec \left (\lambda x \right )}{a \left (c_{3} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )+\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {b a}{\lambda ^{2}}}\, \cos \left (\lambda x \right )^{1+\frac {n}{2}}}{n +2}\right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-a \sin \left (\lambda x \right ) y^{2}=b \sin \left (\lambda x \right ) \cos \left (\lambda x \right )^{n} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=a \sin \left (\lambda x \right ) y^{2}+b \sin \left (\lambda x \right ) \cos \left (\lambda x \right )^{n} \end {array} \]
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 sub-methods: trying Riccati_symmetries trying Riccati to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = lambda*cos(lambda*x)*(diff(y(x), x))/sin(lambda*x)-a*sin(lambda*x)^2*b 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 <- Bessel successful <- special function solution successful Change of variables used: [x = arccos(t)/lambda] Linear ODE actually solved: 4*a*b*t^n*(-t^2+1)^(3/2)*u(t)+4*(-t^2+1)^(3/2)*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: 256
dsolve(diff(y(x),x)=a*sin(lambda*x)*y(x)^2+b*sin(lambda*x)*cos(lambda*x)^n,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-\sqrt {\frac {a b}{\lambda ^{2}}}\, \operatorname {BesselY}\left (\frac {3+n}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (x \lambda \right )^{\frac {n}{2}+1}}{n +2}\right ) \cos \left (x \lambda \right )^{\frac {n}{2}+1} c_{1} -\operatorname {BesselJ}\left (\frac {3+n}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (x \lambda \right )^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (x \lambda \right )^{\frac {n}{2}+1}+\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (x \lambda \right )^{\frac {n}{2}+1}}{n +2}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (x \lambda \right )^{\frac {n}{2}+1}}{n +2}\right )\right ) \sec \left (x \lambda \right ) \lambda }{\left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (x \lambda \right )^{\frac {n}{2}+1}}{n +2}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {\frac {a b}{\lambda ^{2}}}\, \cos \left (x \lambda \right )^{\frac {n}{2}+1}}{n +2}\right )\right ) a} \]
✓ Solution by Mathematica
Time used: 1.409 (sec). Leaf size: 695
DSolve[y'[x]==a*Sin[\[Lambda]*x]*y[x]^2+b*Sin[\[Lambda]*x]*Cos[\[Lambda]*x]^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {a} \sqrt {b} \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \cos ^{\frac {n}{2}}(\lambda x) \operatorname {BesselJ}\left (\frac {1}{n+2}-1,\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )-\sqrt {a} \sqrt {b} \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \cos ^{\frac {n}{2}}(\lambda x) \operatorname {BesselJ}\left (1+\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )+\lambda \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \sec (\lambda x) \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )-\sqrt {a} \sqrt {b} c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \cos ^{\frac {n}{2}}(\lambda x) \operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )+\sqrt {a} \sqrt {b} c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \cos ^{\frac {n}{2}}(\lambda x) \operatorname {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )+c_1 \lambda \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \sec (\lambda x) \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )}{2 a \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )+2 a c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )} \\ y(x)\to \frac {\frac {\sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}}(\lambda x) \left (\operatorname {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )-\operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )}{\operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cos ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )}+\lambda \sec (\lambda x)}{2 a} \\ \end{align*}