Internal problem ID [10538]
Internal file name [OUTPUT/9486_Monday_June_06_2022_02_50_49_PM_24821107/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 1.2.6-4. Equations with
cotangent.
Problem number: 41.
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]
\[ \boxed {y^{\prime }-y^{2}-a \cot \left (\beta x \right ) y=a b \cot \left (\beta x \right )-b^{2}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= y^{2}+a \cot \left (\beta x \right ) y +a b \cot \left (\beta x \right )-b^{2} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = y^{2}+a \cot \left (\beta x \right ) y +a b \cot \left (\beta x \right )-b^{2} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=a b \cot \left (\beta x \right )-b^{2}\), \(f_1(x)=\cot \left (\beta x \right ) a\) 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 &=\cot \left (\beta x \right ) a\\ f_2^2 f_0 &=a b \cot \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 )-\cot \left (\beta x \right ) a u^{\prime }\left (x \right )+\left (a b \cot \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
\[ u \left (x \right ) = -\frac {{\mathrm e}^{b x} \left (i c_{2} \beta \left (\int \sec \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \csc \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \sin \left (\beta x \right )^{\frac {a}{\beta }} \cos \left (\beta x \right ) {\mathrm e}^{-2 b x}d x \right )-2 c_{1} \right )}{2} \] The above shows that \[ u^{\prime }\left (x \right ) = -\frac {i \left (\int \sec \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \csc \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \sin \left (\beta x \right )^{\frac {a}{\beta }} \cos \left (\beta x \right ) {\mathrm e}^{-2 b x}d x \right ) c_{2} b \beta \,{\mathrm e}^{b x}}{2}-\frac {i \cos \left (\beta x \right ) \sin \left (\beta x \right )^{\frac {a}{\beta }} \csc \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \sec \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \beta c_{2} {\mathrm e}^{-b x}}{2}+c_{1} b \,{\mathrm e}^{b x} \] Using the above in (1) gives the solution \[ y = \frac {2 \left (-\frac {i \left (\int \sec \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \csc \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \sin \left (\beta x \right )^{\frac {a}{\beta }} \cos \left (\beta x \right ) {\mathrm e}^{-2 b x}d x \right ) c_{2} b \beta \,{\mathrm e}^{b x}}{2}-\frac {i \cos \left (\beta x \right ) \sin \left (\beta x \right )^{\frac {a}{\beta }} \csc \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \sec \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \beta c_{2} {\mathrm e}^{-b x}}{2}+c_{1} b \,{\mathrm e}^{b x}\right ) {\mathrm e}^{-b x}}{i c_{2} \beta \left (\int \sec \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \csc \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \sin \left (\beta x \right )^{\frac {a}{\beta }} \cos \left (\beta x \right ) {\mathrm e}^{-2 b x}d x \right )-2 c_{1}} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {-2 i b c_{3} -\beta \sec \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \csc \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \sin \left (\beta x \right )^{\frac {a}{\beta }} \cos \left (\beta x \right ) {\mathrm e}^{-2 b x}-\beta \left (\int \sec \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \csc \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \sin \left (\beta x \right )^{\frac {a}{\beta }} \cos \left (\beta x \right ) {\mathrm e}^{-2 b x}d x \right ) b}{\beta \left (\int \sec \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \csc \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \sin \left (\beta x \right )^{\frac {a}{\beta }} \cos \left (\beta x \right ) {\mathrm e}^{-2 b x}d x \right )+2 i c_{3}} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {-2 i b c_{3} -\beta \sec \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \csc \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \sin \left (\beta x \right )^{\frac {a}{\beta }} \cos \left (\beta x \right ) {\mathrm e}^{-2 b x}-\beta \left (\int \sec \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \csc \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \sin \left (\beta x \right )^{\frac {a}{\beta }} \cos \left (\beta x \right ) {\mathrm e}^{-2 b x}d x \right ) b}{\beta \left (\int \sec \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \csc \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \sin \left (\beta x \right )^{\frac {a}{\beta }} \cos \left (\beta x \right ) {\mathrm e}^{-2 b x}d x \right )+2 i c_{3}} \\ \end{align*}
Verification of solutions
\[ y = \frac {-2 i b c_{3} -\beta \sec \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \csc \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \sin \left (\beta x \right )^{\frac {a}{\beta }} \cos \left (\beta x \right ) {\mathrm e}^{-2 b x}-\beta \left (\int \sec \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \csc \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \sin \left (\beta x \right )^{\frac {a}{\beta }} \cos \left (\beta x \right ) {\mathrm e}^{-2 b x}d x \right ) b}{\beta \left (\int \sec \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \csc \left (\frac {\pi }{4}+\frac {\beta x}{2}\right ) \sin \left (\beta x \right )^{\frac {a}{\beta }} \cos \left (\beta x \right ) {\mathrm e}^{-2 b x}d x \right )+2 i c_{3}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2}-a \cot \left (\beta x \right ) y=a b \cot \left (\beta x \right )-b^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{2}+a \cot \left (\beta x \right ) y+a b \cot \left (\beta x \right )-b^{2} \end {array} \]
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*cot(beta*x)*y(x)+a*b*cot(beta*x)-b^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-\left (\csc \left (x \beta \right )^{2}\right )^{-\frac {a}{2 \beta }} {\mathrm e}^{-2 b x}-b \left (\int \left (\csc \left (x \beta \right )^{2}\right )^{-\frac {a}{2 \beta }} {\mathrm e}^{-2 b x}d x -c_{1} \right )}{\int \left (\csc \left (x \beta \right )^{2}\right )^{-\frac {a}{2 \beta }} {\mathrm e}^{-2 b x}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 26.26 (sec). Leaf size: 462
DSolve[y'[x]==y[x]^2+a*Cot[\[Beta]*x]*y[x]+a*b*Cot[\[Beta]*x]-b^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {b (i a+2 b-2 i \beta ) \left (-i e^{-i \beta x} \left (-1+e^{2 i \beta x}\right )\right )^{a/\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 )+(a-2 i b) \left ((a-2 i b-2 \beta ) \left (\left (-i e^{-i \beta x} \left (-1+e^{2 i \beta x}\right )\right )^{a/\beta }-a b \beta c_1 2^{a/\beta } e^{2 b x}\right )-i b e^{2 i \beta x} \left (-i e^{-i \beta x} \left (-1+e^{2 i \beta x}\right )\right )^{a/\beta } \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 ) \left (-i e^{-i \beta x} \left (-1+e^{2 i \beta x}\right )\right )^{a/\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 )+(a-2 i b) \left (i e^{2 i \beta x} \left (-i e^{-i \beta x} \left (-1+e^{2 i \beta x}\right )\right )^{a/\beta } \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 )+a \beta c_1 2^{a/\beta } e^{2 b x} (a-2 i b-2 \beta )\right )} \\ y(x)\to -b \\ \end{align*}