Internal problem ID [10543]
Internal file name [OUTPUT/9491_Monday_June_06_2022_02_53_25_PM_4136660/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: 46.
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]
Unable to solve or complete the solution.
\[ \boxed {\left (\cot \left (\lambda x \right ) a +b \right ) y^{\prime }-y^{2}-c \cot \left (\mu x \right ) y=-d^{2}+c d \cot \left (\mu x \right )} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {y^{2}+c \cot \left (\mu x \right ) y -d^{2}+c d \cot \left (\mu x \right )}{\cot \left (\lambda x \right ) a +b} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \frac {c d \cot \left (\mu x \right )}{\cot \left (\lambda x \right ) a +b}+\frac {c \cot \left (\mu x \right ) y}{\cot \left (\lambda x \right ) a +b}-\frac {d^{2}}{\cot \left (\lambda x \right ) a +b}+\frac {y^{2}}{\cot \left (\lambda x \right ) a +b} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {-d^{2}+c d \cot \left (\mu x \right )}{\cot \left (\lambda x \right ) a +b}\), \(f_1(x)=\frac {c \cot \left (\mu x \right )}{\cot \left (\lambda x \right ) a +b}\) and \(f_2(x)=\frac {1}{\cot \left (\lambda x \right ) a +b}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\frac {u}{\cot \left (\lambda x \right ) a +b}} \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' &=-\frac {\lambda \left (-1-\cot \left (\lambda x \right )^{2}\right ) a}{\left (\cot \left (\lambda x \right ) a +b \right )^{2}}\\ f_1 f_2 &=\frac {c \cot \left (\mu x \right )}{\left (\cot \left (\lambda x \right ) a +b \right )^{2}}\\ f_2^2 f_0 &=\frac {-d^{2}+c d \cot \left (\mu x \right )}{\left (\cot \left (\lambda x \right ) a +b \right )^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \frac {u^{\prime \prime }\left (x \right )}{\cot \left (\lambda x \right ) a +b}-\left (-\frac {\lambda \left (-1-\cot \left (\lambda x \right )^{2}\right ) a}{\left (\cot \left (\lambda x \right ) a +b \right )^{2}}+\frac {c \cot \left (\mu x \right )}{\left (\cot \left (\lambda x \right ) a +b \right )^{2}}\right ) u^{\prime }\left (x \right )+\frac {\left (-d^{2}+c d \cot \left (\mu x \right )\right ) u \left (x \right )}{\left (\cot \left (\lambda x \right ) a +b \right )^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives Unable to solve. Terminating.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\cot \left (\lambda x \right ) a +b \right ) y^{\prime }-y^{2}-c \cot \left (\mu x \right ) y=-d^{2}+c d \cot \left (\mu x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{2}+c \cot \left (\mu x \right ) y-d^{2}+c d \cot \left (\mu x \right )}{\cot \left (\lambda x \right ) a +b} \end {array} \]
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: 0.015 (sec). Leaf size: 366
dsolve((a*cot(lambda*x)+b)*diff(y(x),x)=y(x)^2+c*cot(mu*x)*y(x)-d^2+c*d*cot(mu*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {-{\mathrm e}^{\frac {\lambda c \left (a^{2}+b^{2}\right ) \left (\int \frac {\cot \left (x \mu \right )}{a \cot \left (x \lambda \right )+b}d x \right )-2 d \left (\operatorname {arccot}\left (\cot \left (x \lambda \right )\right )-\frac {\pi }{2}\right ) b}{\lambda \left (a^{2}+b^{2}\right )}} \left (\csc \left (x \lambda \right )^{2}\right )^{-\frac {a d}{\lambda \left (a^{2}+b^{2}\right )}} \left (a \cot \left (x \lambda \right )+b \right )^{\frac {2 a d}{\lambda \left (a^{2}+b^{2}\right )}}-d \left (\int \left (a \cot \left (x \lambda \right )+b \right )^{\frac {\left (-a^{2}-b^{2}\right ) \lambda +2 a d}{\lambda \left (a^{2}+b^{2}\right )}} {\mathrm e}^{\frac {\lambda c \left (a^{2}+b^{2}\right ) \left (\int \frac {\cot \left (x \mu \right )}{a \cot \left (x \lambda \right )+b}d x \right )-2 d \left (\operatorname {arccot}\left (\cot \left (x \lambda \right )\right )-\frac {\pi }{2}\right ) b}{\lambda \left (a^{2}+b^{2}\right )}} \left (\csc \left (x \lambda \right )^{2}\right )^{-\frac {a d}{\lambda \left (a^{2}+b^{2}\right )}}d x -c_{1} \right )}{\int \left (a \cot \left (x \lambda \right )+b \right )^{\frac {\left (-a^{2}-b^{2}\right ) \lambda +2 a d}{\lambda \left (a^{2}+b^{2}\right )}} {\mathrm e}^{\frac {\lambda c \left (a^{2}+b^{2}\right ) \left (\int \frac {\cot \left (x \mu \right )}{a \cot \left (x \lambda \right )+b}d x \right )-2 d \left (\operatorname {arccot}\left (\cot \left (x \lambda \right )\right )-\frac {\pi }{2}\right ) b}{\lambda \left (a^{2}+b^{2}\right )}} \left (\csc \left (x \lambda \right )^{2}\right )^{-\frac {a d}{\lambda \left (a^{2}+b^{2}\right )}}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 87.594 (sec). Leaf size: 799
DSolve[(a*Cot[\[Lambda]*x]+b)*y'[x]==y[x]^2+c*Cot[\[Mu]*x]*y[x]-d^2+c*d*Cot[\[Mu]*x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\frac {e^{-\int _1^{K[2]}-\frac {\csc (\mu K[1]) (-2 d \cos (\lambda K[1]-\mu K[1])+2 d \cos (\lambda K[1]+\mu K[1])+c \sin (\lambda K[1]-\mu K[1])+c \sin (\lambda K[1]+\mu K[1]))}{2 (a \cos (\lambda K[1])+b \sin (\lambda K[1]))}dK[1]} (-d \cos (\lambda K[2]-\mu K[2])+y(x) \cos (\lambda K[2]-\mu K[2])+d \cos (\lambda K[2]+\mu K[2])+c \sin (\lambda K[2]-\mu K[2])+c \sin (\lambda K[2]+\mu K[2])-\cos (\lambda K[2]+\mu K[2]) y(x))}{c \mu (b \cos (\lambda K[2]-\mu K[2])-b \cos (\lambda K[2]+\mu K[2])-a \sin (\lambda K[2]-\mu K[2])+a \sin (\lambda K[2]+\mu K[2])) (d+y(x))}dK[2]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {e^{-\int _1^{K[2]}-\frac {\csc (\mu K[1]) (-2 d \cos (\lambda K[1]-\mu K[1])+2 d \cos (\lambda K[1]+\mu K[1])+c \sin (\lambda K[1]-\mu K[1])+c \sin (\lambda K[1]+\mu K[1]))}{2 (a \cos (\lambda K[1])+b \sin (\lambda K[1]))}dK[1]} (\cos (\lambda K[2]-\mu K[2])-\cos (\lambda K[2]+\mu K[2]))}{c \mu (d+K[3]) (b \cos (\lambda K[2]-\mu K[2])-b \cos (\lambda K[2]+\mu K[2])-a \sin (\lambda K[2]-\mu K[2])+a \sin (\lambda K[2]+\mu K[2]))}-\frac {e^{-\int _1^{K[2]}-\frac {\csc (\mu K[1]) (-2 d \cos (\lambda K[1]-\mu K[1])+2 d \cos (\lambda K[1]+\mu K[1])+c \sin (\lambda K[1]-\mu K[1])+c \sin (\lambda K[1]+\mu K[1]))}{2 (a \cos (\lambda K[1])+b \sin (\lambda K[1]))}dK[1]} (-d \cos (\lambda K[2]-\mu K[2])+K[3] \cos (\lambda K[2]-\mu K[2])+d \cos (\lambda K[2]+\mu K[2])-\cos (\lambda K[2]+\mu K[2]) K[3]+c \sin (\lambda K[2]-\mu K[2])+c \sin (\lambda K[2]+\mu K[2]))}{c \mu (d+K[3])^2 (b \cos (\lambda K[2]-\mu K[2])-b \cos (\lambda K[2]+\mu K[2])-a \sin (\lambda K[2]-\mu K[2])+a \sin (\lambda K[2]+\mu K[2]))}\right )dK[2]-\frac {e^{-\int _1^x-\frac {\csc (\mu K[1]) (-2 d \cos (\lambda K[1]-\mu K[1])+2 d \cos (\lambda K[1]+\mu K[1])+c \sin (\lambda K[1]-\mu K[1])+c \sin (\lambda K[1]+\mu K[1]))}{2 (a \cos (\lambda K[1])+b \sin (\lambda K[1]))}dK[1]}}{c \mu (d+K[3])^2}\right )dK[3]=c_1,y(x)\right ] \]