Internal problem ID [10552]
Internal file name [OUTPUT/9500_Monday_June_06_2022_02_58_37_PM_44320937/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: 55.
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]
\[ \boxed {y^{\prime }-y^{2}-m y \cot \left (x \right )=b^{2} \sin \left (x \right )^{2 m}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= y^{2}+m y \cot \left (x \right )+b^{2} \sin \left (x \right )^{2 m} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = y^{2}+m y \cot \left (x \right )+b^{2} \sin \left (x \right )^{2 m} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=b^{2} \sin \left (x \right )^{2 m}\), \(f_1(x)=m \cot \left (x \right )\) 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 &=m \cot \left (x \right )\\ f_2^2 f_0 &=b^{2} \sin \left (x \right )^{2 m} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} u^{\prime \prime }\left (x \right )-m \cot \left (x \right ) u^{\prime }\left (x \right )+b^{2} \sin \left (x \right )^{2 m} u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = c_{1} \sin \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )+c_{2} \cos \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\left (c_{2} \sin \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )-c_{1} \cos \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )\right ) \left (\csc \left (x \right )^{2}\right )^{-\frac {m}{2}} \left (-\frac {\cot \left (x \right )^{2} \left (2+m \right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}, 2+\frac {m}{2}\right ], \left [\frac {5}{2}\right ], -\cot \left (x \right )^{2}\right )}{3}+\operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right )\right ) b}{\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}} \] Using the above in (1) gives the solution \[ y = -\frac {\left (c_{2} \sin \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )-c_{1} \cos \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )\right ) \left (\csc \left (x \right )^{2}\right )^{-\frac {m}{2}} \left (-\frac {\cot \left (x \right )^{2} \left (2+m \right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}, 2+\frac {m}{2}\right ], \left [\frac {5}{2}\right ], -\cot \left (x \right )^{2}\right )}{3}+\operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right )\right ) b}{\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \left (c_{1} \sin \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )+c_{2} \cos \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {\left (-\sin \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )+c_{3} \cos \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )\right ) \left (\csc \left (x \right )^{2}\right )^{-\frac {m}{2}} \left (-\frac {\cot \left (x \right )^{2} \left (2+m \right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}, 2+\frac {m}{2}\right ], \left [\frac {5}{2}\right ], -\cot \left (x \right )^{2}\right )}{3}+\operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right )\right ) b}{\left (c_{3} \sin \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )+\cos \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )\right ) \sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (-\sin \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )+c_{3} \cos \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )\right ) \left (\csc \left (x \right )^{2}\right )^{-\frac {m}{2}} \left (-\frac {\cot \left (x \right )^{2} \left (2+m \right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}, 2+\frac {m}{2}\right ], \left [\frac {5}{2}\right ], -\cot \left (x \right )^{2}\right )}{3}+\operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right )\right ) b}{\left (c_{3} \sin \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )+\cos \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )\right ) \sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}} \\ \end{align*}
Verification of solutions
\[ y = \frac {\left (-\sin \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )+c_{3} \cos \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )\right ) \left (\csc \left (x \right )^{2}\right )^{-\frac {m}{2}} \left (-\frac {\cot \left (x \right )^{2} \left (2+m \right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}, 2+\frac {m}{2}\right ], \left [\frac {5}{2}\right ], -\cot \left (x \right )^{2}\right )}{3}+\operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right )\right ) b}{\left (c_{3} \sin \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )+\cos \left (\sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )\right ) \sqrt {\sin \left (x \right )^{4} \left (\csc \left (x \right )^{2}\right )^{-m}}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2}-m y \cot \left (x \right )=b^{2} \sin \left (x \right )^{2 m} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{2}+m y \cot \left (x \right )+b^{2} \sin \left (x \right )^{2 m} \end {array} \]
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) = m*cot(x)*(diff(y(x), x))-b^2*sin(x)^(2*m)*y(x), y(x)` *** Subleve 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)] <- linear_1 successful Change of variables used: [x = arccot(t)] Linear ODE actually solved: b^2*u(t)+t*(m*t^2+2*t^2+m+2)*(t^2+1)^m*diff(u(t),t)+(t^4+2*t^2+1)*(t^2+1)^m*diff(diff(u(t),t),t) = 0 <- change of variables successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 281
dsolve(diff(y(x),x)=y(x)^2+m*y(x)*cot(x)+b^2*sin(x)^(2*m),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\sqrt {\left (\csc \left (x \right )^{2}\right )^{-m} \sin \left (x \right )^{4}}\, \left (-c_{1} \sin \left (\sqrt {\left (\csc \left (x \right )^{2}\right )^{-m} \sin \left (x \right )^{4}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )+\cos \left (\sqrt {\left (\csc \left (x \right )^{2}\right )^{-m} \sin \left (x \right )^{4}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )\right ) \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} b \csc \left (x \right )^{6} \left (-\frac {\operatorname {hypergeom}\left (\left [\frac {3}{2}, 2+\frac {m}{2}\right ], \left [\frac {5}{2}\right ], -\cot \left (x \right )^{2}\right ) \cos \left (x \right )^{2} \left (m +2\right )}{3}+\operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) \sin \left (x \right )^{2}\right )}{c_{1} \cos \left (\sqrt {\left (\csc \left (x \right )^{2}\right )^{-m} \sin \left (x \right )^{4}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )+\sin \left (\sqrt {\left (\csc \left (x \right )^{2}\right )^{-m} \sin \left (x \right )^{4}}\, \csc \left (x \right )^{2} \left (\csc \left (x \right )^{2}\right )^{\frac {m}{2}} \cot \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}\right ], -\cot \left (x \right )^{2}\right ) b \right )} \]
✓ Solution by Mathematica
Time used: 5.352 (sec). Leaf size: 72
DSolve[y'[x]==y[x]^2+m*y[x]*Cot[x]+b^2*Sin[x]^(2*m),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \sqrt {b^2} \sin ^m(x) \tan \left (\frac {\sqrt {b^2} \sqrt {\cos ^2(x)} \sec (x) \sin ^{m+1}(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(x)\right )}{m+1}+c_1\right ) \]