Internal problem ID [10551]
Internal file name [OUTPUT/9499_Monday_June_06_2022_02_58_20_PM_18999849/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: 54.
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]
\[ \boxed {y^{\prime }-y^{2}+m y \tan \left (x \right )=b^{2} \cos \left (x \right )^{2 m}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= y^{2}-m y \tan \left (x \right )+b^{2} \cos \left (x \right )^{2 m} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = y^{2}-m y \tan \left (x \right )+b^{2} \cos \left (x \right )^{2 m} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=b^{2} \cos \left (x \right )^{2 m}\), \(f_1(x)=-m \tan \left (x \right )\) 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 \tan \left (x \right )\\ f_2^2 f_0 &=b^{2} \cos \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 \tan \left (x \right ) u^{\prime }\left (x \right )+b^{2} \cos \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
\[ u \left (x \right ) = c_{1} \sin \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )+c_{2} \cos \left (b \sqrt {\cos \left (x \right )^{2 m -2}}\, \cos \left (x \right )^{-m +1} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \left (-\cos \left (x \right ) \sin \left (b \sqrt {\cos \left (x \right )^{2 m -2}}\, \cos \left (x \right )^{-m +1} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \sqrt {\cos \left (x \right )^{2 m -2}}\, c_{2} +\cos \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \sqrt {\cos \left (x \right )^{2 m}}\, c_{1} \right ) \left (-\frac {\sin \left (x \right )^{2} \left (m -1\right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}, -\frac {m}{2}+\frac {3}{2}\right ], \left [\frac {5}{2}\right ], \sin \left (x \right )^{2}\right )}{3}+\operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \cos \left (x \right )^{-m +1} b \] Using the above in (1) gives the solution \[ y = -\frac {\left (-\cos \left (x \right ) \sin \left (b \sqrt {\cos \left (x \right )^{2 m -2}}\, \cos \left (x \right )^{-m +1} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \sqrt {\cos \left (x \right )^{2 m -2}}\, c_{2} +\cos \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \sqrt {\cos \left (x \right )^{2 m}}\, c_{1} \right ) \left (-\frac {\sin \left (x \right )^{2} \left (m -1\right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}, -\frac {m}{2}+\frac {3}{2}\right ], \left [\frac {5}{2}\right ], \sin \left (x \right )^{2}\right )}{3}+\operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \cos \left (x \right )^{-m +1} b}{c_{1} \sin \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )+c_{2} \cos \left (b \sqrt {\cos \left (x \right )^{2 m -2}}\, \cos \left (x \right )^{-m +1} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {\left (\sin \left (x \right )^{2} \left (m -1\right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}, -\frac {m}{2}+\frac {3}{2}\right ], \left [\frac {5}{2}\right ], \sin \left (x \right )^{2}\right )-3 \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \left (\cos \left (x \right ) \sin \left (b \sqrt {\cos \left (x \right )^{2 m -2}}\, \cos \left (x \right )^{-m +1} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \sqrt {\cos \left (x \right )^{2 m -2}}-\cos \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \sqrt {\cos \left (x \right )^{2 m}}\, c_{3} \right ) \cos \left (x \right )^{-m +1} b}{3 c_{3} \sin \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )+3 \cos \left (b \sqrt {\cos \left (x \right )^{2 m -2}}\, \cos \left (x \right )^{-m +1} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\left (\sin \left (x \right )^{2} \left (m -1\right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}, -\frac {m}{2}+\frac {3}{2}\right ], \left [\frac {5}{2}\right ], \sin \left (x \right )^{2}\right )-3 \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \left (\cos \left (x \right ) \sin \left (b \sqrt {\cos \left (x \right )^{2 m -2}}\, \cos \left (x \right )^{-m +1} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \sqrt {\cos \left (x \right )^{2 m -2}}-\cos \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \sqrt {\cos \left (x \right )^{2 m}}\, c_{3} \right ) \cos \left (x \right )^{-m +1} b}{3 c_{3} \sin \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )+3 \cos \left (b \sqrt {\cos \left (x \right )^{2 m -2}}\, \cos \left (x \right )^{-m +1} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\left (\sin \left (x \right )^{2} \left (m -1\right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}, -\frac {m}{2}+\frac {3}{2}\right ], \left [\frac {5}{2}\right ], \sin \left (x \right )^{2}\right )-3 \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \left (\cos \left (x \right ) \sin \left (b \sqrt {\cos \left (x \right )^{2 m -2}}\, \cos \left (x \right )^{-m +1} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \sqrt {\cos \left (x \right )^{2 m -2}}-\cos \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \sqrt {\cos \left (x \right )^{2 m}}\, c_{3} \right ) \cos \left (x \right )^{-m +1} b}{3 c_{3} \sin \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )+3 \cos \left (b \sqrt {\cos \left (x \right )^{2 m -2}}\, \cos \left (x \right )^{-m +1} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2}+m y \tan \left (x \right )=b^{2} \cos \left (x \right )^{2 m} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{2}-m y \tan \left (x \right )+b^{2} \cos \left (x \right )^{2 m} \end {array} \]
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*tan(x)*(diff(y(x), x))-b^2*cos(x)^(2*m)*y(x), y(x)` *** Sublev 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 = arcsin(t)] Linear ODE actually solved: b^2*(-t^2+1)^m*u(t)+(m*t-t)*diff(u(t),t)+(-t^2+1)*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: 222
dsolve(diff(y(x),x)=y(x)^2-m*y(x)*tan(x)+b^2*cos(x)^(2*m),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-\frac {\sin \left (x \right )^{2} \left (m -1\right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}, -\frac {m}{2}+\frac {3}{2}\right ], \left [\frac {5}{2}\right ], \sin \left (x \right )^{2}\right )}{3}+\operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) b \cos \left (x \right )^{-m +1} \left (\sin \left (b \sqrt {\cos \left (x \right )^{2 m -2}}\, \cos \left (x \right )^{-m +1} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right ) \sqrt {\cos \left (x \right )^{2 m -2}}\, \cos \left (x \right ) c_{1} -\sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )\right )}{c_{1} \cos \left (b \sqrt {\cos \left (x \right )^{2 m -2}}\, \cos \left (x \right )^{-m +1} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )+\sin \left (b \sqrt {\cos \left (x \right )^{2 m}}\, \cos \left (x \right )^{-m} \sin \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, -\frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \sin \left (x \right )^{2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 4.179 (sec). Leaf size: 73
DSolve[y'[x]==y[x]^2-m*y[x]*Tan[x]+b^2*Cos[x]^(2*m),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \sqrt {b^2} \cos ^m(x) \tan \left (-\frac {\sqrt {b^2} \sqrt {\sin ^2(x)} \csc (x) \cos ^{m+1}(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(x)\right )}{m+1}+c_1\right ) \]