Internal problem ID [10528]
Internal file name [OUTPUT/9476_Monday_June_06_2022_02_47_18_PM_75210480/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 1.2.6-3. Equations with
tangent.
Problem number: 31.
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]
\[ \boxed {y^{\prime }-y^{2}-a \tan \left (\beta x \right ) y=a b \tan \left (\beta x \right )-b^{2}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= y^{2}+a \tan \left (\beta x \right ) y +a b \tan \left (\beta x \right )-b^{2} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = y^{2}+a \tan \left (\beta x \right ) y +a b \tan \left (\beta x \right )-b^{2} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=a b \tan \left (\beta x \right )-b^{2}\), \(f_1(x)=\tan \left (\beta x \right ) a\) 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 &=\tan \left (\beta x \right ) a\\ f_2^2 f_0 &=a b \tan \left (\beta x \right )-b^{2} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} u^{\prime \prime }\left (x \right )-\tan \left (\beta x \right ) a u^{\prime }\left (x \right )+\left (a b \tan \left (\beta x \right )-b^{2}\right ) u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = c_{1} \left (\tan \left (\beta x \right )+i\right )^{\frac {i b}{2 \beta }} \left (\tan \left (\beta x \right )-i\right )^{-\frac {i b}{2 \beta }}+c_{2} \left (\tan \left (\beta x \right )+i\right )^{\frac {-i b +a}{2 \beta }} \operatorname {hypergeom}\left (\left [1, \frac {a}{\beta }\right ], \left [\frac {-2 i b +a +2 \beta }{2 \beta }\right ], \frac {1}{2}-\frac {i \tan \left (\beta x \right )}{2}\right ) \left (\tan \left (\beta x \right )-i\right )^{\frac {i b +a}{2 \beta }} \] The above shows that \[ u^{\prime }\left (x \right ) = \left (\frac {c_{1} b \left (\tan \left (\beta x \right )-i\right )^{-\frac {i b}{2 \beta }} \left (\tan \left (\beta x \right )+i\right )^{\frac {i b}{2 \beta }}}{2-2 i \tan \left (\beta x \right )}-\frac {i \left (\tan \left (\beta x \right )-i\right )^{-\frac {i b +2 \beta }{2 \beta }} c_{1} b \left (\tan \left (\beta x \right )+i\right )^{\frac {i b}{2 \beta }}}{2}+\frac {\left (-i b +a \right ) c_{2} \left (\tan \left (\beta x \right )-i\right )^{\frac {i b +a}{2 \beta }} \left (\tan \left (\beta x \right )+i\right )^{\frac {-i b +a -2 \beta }{2 \beta }} \operatorname {hypergeom}\left (\left [1, \frac {a}{\beta }\right ], \left [\frac {-2 i b +a +2 \beta }{2 \beta }\right ], \frac {1}{2}-\frac {i \tan \left (\beta x \right )}{2}\right )}{2}+\frac {c_{2} \left (\tan \left (\beta x \right )+i\right )^{\frac {-i b +a}{2 \beta }} a \operatorname {hypergeom}\left (\left [2, \frac {a +\beta }{\beta }\right ], \left [\frac {-2 i b +a +4 \beta }{2 \beta }\right ], \frac {1}{2}-\frac {i \tan \left (\beta x \right )}{2}\right ) \beta \left (\tan \left (\beta x \right )-i\right )^{\frac {i b +a}{2 \beta }}}{i a +2 i \beta +2 b}+\frac {\left (\tan \left (\beta x \right )-i\right )^{\frac {i b +a -2 \beta }{2 \beta }} c_{2} \left (\tan \left (\beta x \right )+i\right )^{\frac {-i b +a}{2 \beta }} \operatorname {hypergeom}\left (\left [1, \frac {a}{\beta }\right ], \left [\frac {-2 i b +a +2 \beta }{2 \beta }\right ], \frac {1}{2}-\frac {i \tan \left (\beta x \right )}{2}\right ) \left (i b +a \right )}{2}\right ) \left (1+\tan \left (\beta x \right )^{2}\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\left (\frac {c_{1} b \left (\tan \left (\beta x \right )-i\right )^{-\frac {i b}{2 \beta }} \left (\tan \left (\beta x \right )+i\right )^{\frac {i b}{2 \beta }}}{2-2 i \tan \left (\beta x \right )}-\frac {i \left (\tan \left (\beta x \right )-i\right )^{-\frac {i b +2 \beta }{2 \beta }} c_{1} b \left (\tan \left (\beta x \right )+i\right )^{\frac {i b}{2 \beta }}}{2}+\frac {\left (-i b +a \right ) c_{2} \left (\tan \left (\beta x \right )-i\right )^{\frac {i b +a}{2 \beta }} \left (\tan \left (\beta x \right )+i\right )^{\frac {-i b +a -2 \beta }{2 \beta }} \operatorname {hypergeom}\left (\left [1, \frac {a}{\beta }\right ], \left [\frac {-2 i b +a +2 \beta }{2 \beta }\right ], \frac {1}{2}-\frac {i \tan \left (\beta x \right )}{2}\right )}{2}+\frac {c_{2} \left (\tan \left (\beta x \right )+i\right )^{\frac {-i b +a}{2 \beta }} a \operatorname {hypergeom}\left (\left [2, \frac {a +\beta }{\beta }\right ], \left [\frac {-2 i b +a +4 \beta }{2 \beta }\right ], \frac {1}{2}-\frac {i \tan \left (\beta x \right )}{2}\right ) \beta \left (\tan \left (\beta x \right )-i\right )^{\frac {i b +a}{2 \beta }}}{i a +2 i \beta +2 b}+\frac {\left (\tan \left (\beta x \right )-i\right )^{\frac {i b +a -2 \beta }{2 \beta }} c_{2} \left (\tan \left (\beta x \right )+i\right )^{\frac {-i b +a}{2 \beta }} \operatorname {hypergeom}\left (\left [1, \frac {a}{\beta }\right ], \left [\frac {-2 i b +a +2 \beta }{2 \beta }\right ], \frac {1}{2}-\frac {i \tan \left (\beta x \right )}{2}\right ) \left (i b +a \right )}{2}\right ) \left (1+\tan \left (\beta x \right )^{2}\right )}{c_{1} \left (\tan \left (\beta x \right )+i\right )^{\frac {i b}{2 \beta }} \left (\tan \left (\beta x \right )-i\right )^{-\frac {i b}{2 \beta }}+c_{2} \left (\tan \left (\beta x \right )+i\right )^{\frac {-i b +a}{2 \beta }} \operatorname {hypergeom}\left (\left [1, \frac {a}{\beta }\right ], \left [\frac {-2 i b +a +2 \beta }{2 \beta }\right ], \frac {1}{2}-\frac {i \tan \left (\beta x \right )}{2}\right ) \left (\tan \left (\beta x \right )-i\right )^{\frac {i b +a}{2 \beta }}} \] 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 (1+\tan \left (\beta x \right )^{2}\right ) \left (2 a \left (\tan \left (\beta x \right )-i\right )^{\frac {i b +a}{2 \beta }} \beta \left (\tan \left (\beta x \right )+i\right )^{\frac {-i b +a}{2 \beta }} \operatorname {hypergeom}\left (\left [2, \frac {a +\beta }{\beta }\right ], \left [\frac {-2 i b +a +4 \beta }{2 \beta }\right ], \frac {1}{2}-\frac {i \tan \left (\beta x \right )}{2}\right ) \left (\tan \left (\beta x \right )+i\right )+\left (\left (\left (-2 i b^{2}-\left (-3 a -2 \beta \right ) b +i a \left (a +2 \beta \right )\right ) \tan \left (\beta x \right )+2 b^{2}+i b \left (3 a +2 \beta \right )-a \left (a +2 \beta \right )\right ) \left (\tan \left (\beta x \right )-i\right )^{\frac {i b +a}{2 \beta }} \left (\tan \left (\beta x \right )+i\right )^{\frac {-i b +a -2 \beta }{2 \beta }}+\left (\left (2 i b^{2}-\left (-a +2 \beta \right ) b +i a \left (a +2 \beta \right )\right ) \tan \left (\beta x \right )-2 b^{2}+i \left (a -2 \beta \right ) b -a \left (a +2 \beta \right )\right ) \left (\tan \left (\beta x \right )+i\right )^{\frac {-i b +a}{2 \beta }} \left (\tan \left (\beta x \right )-i\right )^{\frac {i b +a -2 \beta }{2 \beta }}\right ) \operatorname {hypergeom}\left (\left [1, \frac {a}{\beta }\right ], \left [\frac {-2 i b +a +2 \beta }{2 \beta }\right ], \frac {1}{2}-\frac {i \tan \left (\beta x \right )}{2}\right )+b \left (\left (\tan \left (\beta x \right )-i\right )^{-\frac {i b +2 \beta }{2 \beta }} \left (\tan \left (\beta x \right ) \left (-2 i b +a +2 \beta \right )+2 b +i a +2 i \beta \right )+\left (2 i b -a -2 \beta \right ) \left (\tan \left (\beta x \right )-i\right )^{-\frac {i b}{2 \beta }}\right ) c_{3} \left (\tan \left (\beta x \right )+i\right )^{\frac {i b}{2 \beta }}\right )}{4 \left (1-i \tan \left (\beta x \right )\right ) \left (c_{3} \left (\tan \left (\beta x \right )+i\right )^{\frac {i b}{2 \beta }} \left (\tan \left (\beta x \right )-i\right )^{-\frac {i b}{2 \beta }}+\left (\tan \left (\beta x \right )+i\right )^{\frac {-i b +a}{2 \beta }} \operatorname {hypergeom}\left (\left [1, \frac {a}{\beta }\right ], \left [\frac {-2 i b +a +2 \beta }{2 \beta }\right ], \frac {1}{2}-\frac {i \tan \left (\beta x \right )}{2}\right ) \left (\tan \left (\beta x \right )-i\right )^{\frac {i b +a}{2 \beta }}\right ) \left (-i b +\frac {a}{2}+\beta \right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (1+\tan \left (\beta x \right )^{2}\right ) \left (2 a \left (\tan \left (\beta x \right )-i\right )^{\frac {i b +a}{2 \beta }} \beta \left (\tan \left (\beta x \right )+i\right )^{\frac {-i b +a}{2 \beta }} \operatorname {hypergeom}\left (\left [2, \frac {a +\beta }{\beta }\right ], \left [\frac {-2 i b +a +4 \beta }{2 \beta }\right ], \frac {1}{2}-\frac {i \tan \left (\beta x \right )}{2}\right ) \left (\tan \left (\beta x \right )+i\right )+\left (\left (\left (-2 i b^{2}-\left (-3 a -2 \beta \right ) b +i a \left (a +2 \beta \right )\right ) \tan \left (\beta x \right )+2 b^{2}+i b \left (3 a +2 \beta \right )-a \left (a +2 \beta \right )\right ) \left (\tan \left (\beta x \right )-i\right )^{\frac {i b +a}{2 \beta }} \left (\tan \left (\beta x \right )+i\right )^{\frac {-i b +a -2 \beta }{2 \beta }}+\left (\left (2 i b^{2}-\left (-a +2 \beta \right ) b +i a \left (a +2 \beta \right )\right ) \tan \left (\beta x \right )-2 b^{2}+i \left (a -2 \beta \right ) b -a \left (a +2 \beta \right )\right ) \left (\tan \left (\beta x \right )+i\right )^{\frac {-i b +a}{2 \beta }} \left (\tan \left (\beta x \right )-i\right )^{\frac {i b +a -2 \beta }{2 \beta }}\right ) \operatorname {hypergeom}\left (\left [1, \frac {a}{\beta }\right ], \left [\frac {-2 i b +a +2 \beta }{2 \beta }\right ], \frac {1}{2}-\frac {i \tan \left (\beta x \right )}{2}\right )+b \left (\left (\tan \left (\beta x \right )-i\right )^{-\frac {i b +2 \beta }{2 \beta }} \left (\tan \left (\beta x \right ) \left (-2 i b +a +2 \beta \right )+2 b +i a +2 i \beta \right )+\left (2 i b -a -2 \beta \right ) \left (\tan \left (\beta x \right )-i\right )^{-\frac {i b}{2 \beta }}\right ) c_{3} \left (\tan \left (\beta x \right )+i\right )^{\frac {i b}{2 \beta }}\right )}{4 \left (1-i \tan \left (\beta x \right )\right ) \left (c_{3} \left (\tan \left (\beta x \right )+i\right )^{\frac {i b}{2 \beta }} \left (\tan \left (\beta x \right )-i\right )^{-\frac {i b}{2 \beta }}+\left (\tan \left (\beta x \right )+i\right )^{\frac {-i b +a}{2 \beta }} \operatorname {hypergeom}\left (\left [1, \frac {a}{\beta }\right ], \left [\frac {-2 i b +a +2 \beta }{2 \beta }\right ], \frac {1}{2}-\frac {i \tan \left (\beta x \right )}{2}\right ) \left (\tan \left (\beta x \right )-i\right )^{\frac {i b +a}{2 \beta }}\right ) \left (-i b +\frac {a}{2}+\beta \right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {\left (1+\tan \left (\beta x \right )^{2}\right ) \left (2 a \left (\tan \left (\beta x \right )-i\right )^{\frac {i b +a}{2 \beta }} \beta \left (\tan \left (\beta x \right )+i\right )^{\frac {-i b +a}{2 \beta }} \operatorname {hypergeom}\left (\left [2, \frac {a +\beta }{\beta }\right ], \left [\frac {-2 i b +a +4 \beta }{2 \beta }\right ], \frac {1}{2}-\frac {i \tan \left (\beta x \right )}{2}\right ) \left (\tan \left (\beta x \right )+i\right )+\left (\left (\left (-2 i b^{2}-\left (-3 a -2 \beta \right ) b +i a \left (a +2 \beta \right )\right ) \tan \left (\beta x \right )+2 b^{2}+i b \left (3 a +2 \beta \right )-a \left (a +2 \beta \right )\right ) \left (\tan \left (\beta x \right )-i\right )^{\frac {i b +a}{2 \beta }} \left (\tan \left (\beta x \right )+i\right )^{\frac {-i b +a -2 \beta }{2 \beta }}+\left (\left (2 i b^{2}-\left (-a +2 \beta \right ) b +i a \left (a +2 \beta \right )\right ) \tan \left (\beta x \right )-2 b^{2}+i \left (a -2 \beta \right ) b -a \left (a +2 \beta \right )\right ) \left (\tan \left (\beta x \right )+i\right )^{\frac {-i b +a}{2 \beta }} \left (\tan \left (\beta x \right )-i\right )^{\frac {i b +a -2 \beta }{2 \beta }}\right ) \operatorname {hypergeom}\left (\left [1, \frac {a}{\beta }\right ], \left [\frac {-2 i b +a +2 \beta }{2 \beta }\right ], \frac {1}{2}-\frac {i \tan \left (\beta x \right )}{2}\right )+b \left (\left (\tan \left (\beta x \right )-i\right )^{-\frac {i b +2 \beta }{2 \beta }} \left (\tan \left (\beta x \right ) \left (-2 i b +a +2 \beta \right )+2 b +i a +2 i \beta \right )+\left (2 i b -a -2 \beta \right ) \left (\tan \left (\beta x \right )-i\right )^{-\frac {i b}{2 \beta }}\right ) c_{3} \left (\tan \left (\beta x \right )+i\right )^{\frac {i b}{2 \beta }}\right )}{4 \left (1-i \tan \left (\beta x \right )\right ) \left (c_{3} \left (\tan \left (\beta x \right )+i\right )^{\frac {i b}{2 \beta }} \left (\tan \left (\beta x \right )-i\right )^{-\frac {i b}{2 \beta }}+\left (\tan \left (\beta x \right )+i\right )^{\frac {-i b +a}{2 \beta }} \operatorname {hypergeom}\left (\left [1, \frac {a}{\beta }\right ], \left [\frac {-2 i b +a +2 \beta }{2 \beta }\right ], \frac {1}{2}-\frac {i \tan \left (\beta x \right )}{2}\right ) \left (\tan \left (\beta x \right )-i\right )^{\frac {i b +a}{2 \beta }}\right ) \left (-i b +\frac {a}{2}+\beta \right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2}-a \tan \left (\beta x \right ) y=a b \tan \left (\beta x \right )-b^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{2}+a \tan \left (\beta x \right ) y+a b \tan \left (\beta x \right )-b^{2} \end {array} \]
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: <- Riccati particular case Kamke (b) successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 81
dsolve(diff(y(x),x)=y(x)^2+a*tan(beta*x)*y(x)+a*b*tan(beta*x)-b^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-\left (\sec \left (x \beta \right )^{2}\right )^{\frac {a}{2 \beta }} {\mathrm e}^{-2 b x}-b \left (\int \left (\sec \left (x \beta \right )^{2}\right )^{\frac {a}{2 \beta }} {\mathrm e}^{-2 b x}d x -c_{1} \right )}{\int \left (\sec \left (x \beta \right )^{2}\right )^{\frac {a}{2 \beta }} {\mathrm e}^{-2 b x}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 25.611 (sec). Leaf size: 408
DSolve[y'[x]==y[x]^2+a*Tan[\[Beta]*x]*y[x]+a*b*Tan[\[Beta]*x]-b^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2^{-\frac {a}{\beta }} \cos ^{-\frac {a}{\beta }}(\beta x) \left (i b (a+2 i b+2 \beta ) \operatorname {Hypergeometric2F1}\left (1,-\frac {a-2 i b}{2 \beta },\frac {a+2 i b+2 \beta }{2 \beta },-e^{2 i x \beta }\right ) (\sin (2 \beta x) \csc (\beta x))^{a/\beta }-(a+2 i b) \left ((a+2 i b+2 \beta ) \left (e^{-i \beta x}+e^{i \beta x}\right )^{a/\beta } \left (1+a b \beta c_1 e^{2 b x} \cos ^{\frac {a}{\beta }}(\beta x)\right )-i b e^{2 i \beta x} \operatorname {Hypergeometric2F1}\left (1,-\frac {a-2 i b-2 \beta }{2 \beta },\frac {a+2 i b+4 \beta }{2 \beta },-e^{2 i x \beta }\right ) (\sin (2 \beta x) \csc (\beta x))^{a/\beta }\right )\right )}{(a+2 i b) \left (a \beta c_1 e^{2 b x} (a+2 i b+2 \beta ) \cos ^{\frac {a}{\beta }}(\beta x)-i e^{2 i \beta x} \operatorname {Hypergeometric2F1}\left (1,-\frac {a-2 i b-2 \beta }{2 \beta },\frac {a+2 i b+4 \beta }{2 \beta },-e^{2 i x \beta }\right )\right )-i (a+2 i b+2 \beta ) \operatorname {Hypergeometric2F1}\left (1,-\frac {a-2 i b}{2 \beta },\frac {a+2 i b+2 \beta }{2 \beta },-e^{2 i x \beta }\right )} \\ y(x)\to -b \\ \end{align*}