Internal problem ID [10472]
Internal file name [OUTPUT/9420_Monday_June_06_2022_02_30_03_PM_79434428/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 1.2.4-2. Equations with hyperbolic
tangent and cotangent.
Problem number: 25.
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.
\[ \boxed {\left (a \coth \left (\lambda x \right )+b \right ) y^{\prime }-y^{2}-c \coth \left (\mu x \right ) y=-d^{2}+c d \coth \left (\mu x \right )} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {y^{2}+c \coth \left (\mu x \right ) y -d^{2}+c d \coth \left (\mu x \right )}{a \coth \left (\lambda x \right )+b} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \frac {c d \coth \left (\mu x \right )}{a \coth \left (\lambda x \right )+b}+\frac {c \coth \left (\mu x \right ) y}{a \coth \left (\lambda x \right )+b}-\frac {d^{2}}{a \coth \left (\lambda x \right )+b}+\frac {y^{2}}{a \coth \left (\lambda x \right )+b} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {-d^{2}+c d \coth \left (\mu x \right )}{a \coth \left (\lambda x \right )+b}\), \(f_1(x)=\frac {c \coth \left (\mu x \right )}{a \coth \left (\lambda x \right )+b}\) and \(f_2(x)=\frac {1}{a \coth \left (\lambda x \right )+b}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\frac {u}{a \coth \left (\lambda x \right )+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 {a \lambda \left (1-\coth \left (\lambda x \right )^{2}\right )}{\left (a \coth \left (\lambda x \right )+b \right )^{2}}\\ f_1 f_2 &=\frac {c \coth \left (\mu x \right )}{\left (a \coth \left (\lambda x \right )+b \right )^{2}}\\ f_2^2 f_0 &=\frac {-d^{2}+c d \coth \left (\mu x \right )}{\left (a \coth \left (\lambda x \right )+b \right )^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \frac {u^{\prime \prime }\left (x \right )}{a \coth \left (\lambda x \right )+b}-\left (-\frac {a \lambda \left (1-\coth \left (\lambda x \right )^{2}\right )}{\left (a \coth \left (\lambda x \right )+b \right )^{2}}+\frac {c \coth \left (\mu x \right )}{\left (a \coth \left (\lambda x \right )+b \right )^{2}}\right ) u^{\prime }\left (x \right )+\frac {\left (-d^{2}+c d \coth \left (\mu x \right )\right ) u \left (x \right )}{\left (a \coth \left (\lambda x \right )+b \right )^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives Unable to solve. Terminating.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (a \coth \left (\lambda x \right )+b \right ) y^{\prime }-y^{2}-c \coth \left (\mu x \right ) y=-d^{2}+c d \coth \left (\mu x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{2}+c \coth \left (\mu x \right ) y-d^{2}+c d \coth \left (\mu x \right )}{a \coth \left (\lambda x \right )+b} \end {array} \]
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: 0.031 (sec). Leaf size: 302
dsolve((a*coth(lambda*x)+b)*diff(y(x),x)=y(x)^2+c*coth(mu*x)*y(x)-d^2+c*d*coth(mu*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {-{\mathrm e}^{c \left (\int \frac {\coth \left (x \mu \right )}{a \coth \left (x \lambda \right )+b}d x \right )} \left (\coth \left (x \lambda \right )+1\right )^{\frac {d}{\lambda \left (a -b \right )}} \left (\coth \left (x \lambda \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (a \coth \left (x \lambda \right )+b \right )^{-\frac {2 a d}{\lambda \left (a^{2}-b^{2}\right )}}-d \left (\int \left (a \coth \left (x \lambda \right )+b \right )^{\frac {\left (-a^{2}+b^{2}\right ) \lambda -2 a d}{\lambda \left (a^{2}-b^{2}\right )}} \left (\coth \left (x \lambda \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (\coth \left (x \lambda \right )+1\right )^{\frac {d}{\lambda \left (a -b \right )}} {\mathrm e}^{c \left (\int \frac {\coth \left (x \mu \right )}{a \coth \left (x \lambda \right )+b}d x \right )}d x -c_{1} \right )}{\int \left (a \coth \left (x \lambda \right )+b \right )^{\frac {\left (-a^{2}+b^{2}\right ) \lambda -2 a d}{\lambda \left (a^{2}-b^{2}\right )}} \left (\coth \left (x \lambda \right )-1\right )^{\frac {d}{\lambda \left (a +b \right )}} \left (\coth \left (x \lambda \right )+1\right )^{\frac {d}{\lambda \left (a -b \right )}} {\mathrm e}^{c \left (\int \frac {\coth \left (x \mu \right )}{a \coth \left (x \lambda \right )+b}d x \right )}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 153.106 (sec). Leaf size: 808
DSolve[(a*Coth[\[Lambda]*x]+b)*y'[x]==y[x]^2+c*Coth[\[Mu]*x]*y[x]-d^2+c*d*Coth[\[Mu]*x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x-\frac {e^{-\int _1^{K[2]}\frac {\text {csch}(\mu K[1]) (-2 d \cosh (\lambda K[1]-\mu K[1])+2 d \cosh (\lambda K[1]+\mu K[1])-c \sinh (\lambda K[1]-\mu K[1])-c \sinh (\lambda K[1]+\mu K[1]))}{2 (a \cosh (\lambda K[1])+b \sinh (\lambda K[1]))}dK[1]} (d \cosh (\lambda K[2]-\mu K[2])-y(x) \cosh (\lambda K[2]-\mu K[2])-d \cosh (\lambda K[2]+\mu K[2])+c \sinh (\lambda K[2]-\mu K[2])+c \sinh (\lambda K[2]+\mu K[2])+\cosh (\lambda K[2]+\mu K[2]) y(x))}{c \mu (b \cosh (\lambda K[2]-\mu K[2])-b \cosh (\lambda K[2]+\mu K[2])+a \sinh (\lambda K[2]-\mu K[2])-a \sinh (\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 {\text {csch}(\mu K[1]) (-2 d \cosh (\lambda K[1]-\mu K[1])+2 d \cosh (\lambda K[1]+\mu K[1])-c \sinh (\lambda K[1]-\mu K[1])-c \sinh (\lambda K[1]+\mu K[1]))}{2 (a \cosh (\lambda K[1])+b \sinh (\lambda K[1]))}dK[1]} (d \cosh (\lambda K[2]-\mu K[2])-K[3] \cosh (\lambda K[2]-\mu K[2])-d \cosh (\lambda K[2]+\mu K[2])+\cosh (\lambda K[2]+\mu K[2]) K[3]+c \sinh (\lambda K[2]-\mu K[2])+c \sinh (\lambda K[2]+\mu K[2]))}{c \mu (d+K[3])^2 (b \cosh (\lambda K[2]-\mu K[2])-b \cosh (\lambda K[2]+\mu K[2])+a \sinh (\lambda K[2]-\mu K[2])-a \sinh (\lambda K[2]+\mu K[2]))}-\frac {e^{-\int _1^{K[2]}\frac {\text {csch}(\mu K[1]) (-2 d \cosh (\lambda K[1]-\mu K[1])+2 d \cosh (\lambda K[1]+\mu K[1])-c \sinh (\lambda K[1]-\mu K[1])-c \sinh (\lambda K[1]+\mu K[1]))}{2 (a \cosh (\lambda K[1])+b \sinh (\lambda K[1]))}dK[1]} (\cosh (\lambda K[2]+\mu K[2])-\cosh (\lambda K[2]-\mu K[2]))}{c \mu (d+K[3]) (b \cosh (\lambda K[2]-\mu K[2])-b \cosh (\lambda K[2]+\mu K[2])+a \sinh (\lambda K[2]-\mu K[2])-a \sinh (\lambda K[2]+\mu K[2]))}\right )dK[2]-\frac {e^{-\int _1^x\frac {\text {csch}(\mu K[1]) (-2 d \cosh (\lambda K[1]-\mu K[1])+2 d \cosh (\lambda K[1]+\mu K[1])-c \sinh (\lambda K[1]-\mu K[1])-c \sinh (\lambda K[1]+\mu K[1]))}{2 (a \cosh (\lambda K[1])+b \sinh (\lambda K[1]))}dK[1]}}{c \mu (d+K[3])^2}\right )dK[3]=c_1,y(x)\right ] \]