Internal problem ID [10456]
Internal file name [OUTPUT/9404_Monday_June_06_2022_02_24_28_PM_71061260/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 1.2.4-1. Equations with hyperbolic
sine and cosine
Problem number: 9.
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]
\[ \boxed {y^{\prime }-y^{2}-a \cosh \left (\beta x \right ) y=a b \cosh \left (\beta x \right )-b^{2}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= y^{2}+a \cosh \left (\beta x \right ) y +a b \cosh \left (\beta x \right )-b^{2} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = y^{2}+a \cosh \left (\beta x \right ) y +a b \cosh \left (\beta x \right )-b^{2} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=a b \cosh \left (\beta x \right )-b^{2}\), \(f_1(x)=a \cosh \left (\beta x \right )\) 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 &=a \cosh \left (\beta x \right )\\ f_2^2 f_0 &=a b \cosh \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 )-a \cosh \left (\beta x \right ) u^{\prime }\left (x \right )+\left (a b \cosh \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
\[ u \left (x \right ) = c_{1} \operatorname {HeunD}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )+c_{2} \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) {\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }} \] The above shows that \[ u^{\prime }\left (x \right ) = -\frac {\operatorname {csch}\left (\frac {\beta x}{2}\right )^{2} c_{1} \beta \operatorname {HeunDPrime}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )}{2}+{\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }} c_{2} \left (-\frac {\operatorname {csch}\left (\frac {\beta x}{2}\right )^{2} \operatorname {HeunDPrime}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) \beta }{2}+a \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) \cosh \left (\beta x \right )\right ) \] Using the above in (1) gives the solution \[ y = -\frac {-\frac {\operatorname {csch}\left (\frac {\beta x}{2}\right )^{2} c_{1} \beta \operatorname {HeunDPrime}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )}{2}+{\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }} c_{2} \left (-\frac {\operatorname {csch}\left (\frac {\beta x}{2}\right )^{2} \operatorname {HeunDPrime}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) \beta }{2}+a \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) \cosh \left (\beta x \right )\right )}{c_{1} \operatorname {HeunD}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )+c_{2} \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) {\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }}} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {-2 \,{\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }} a \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) \cosh \left (\beta x \right )+\beta \operatorname {csch}\left (\frac {\beta x}{2}\right )^{2} \left ({\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }} \operatorname {HeunDPrime}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )+\operatorname {HeunDPrime}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) c_{3} \right )}{2 \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) {\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }}+2 c_{3} \operatorname {HeunD}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {-2 \,{\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }} a \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) \cosh \left (\beta x \right )+\beta \operatorname {csch}\left (\frac {\beta x}{2}\right )^{2} \left ({\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }} \operatorname {HeunDPrime}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )+\operatorname {HeunDPrime}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) c_{3} \right )}{2 \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) {\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }}+2 c_{3} \operatorname {HeunD}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {-2 \,{\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }} a \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) \cosh \left (\beta x \right )+\beta \operatorname {csch}\left (\frac {\beta x}{2}\right )^{2} \left ({\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }} \operatorname {HeunDPrime}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )+\operatorname {HeunDPrime}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) c_{3} \right )}{2 \operatorname {HeunD}\left (\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right ) {\mathrm e}^{\frac {a \sinh \left (\beta x \right )}{\beta }}+2 c_{3} \operatorname {HeunD}\left (-\frac {2 a}{\beta }, \frac {4 b \left (a -b \right )}{\beta ^{2}}, \frac {4 a}{\beta }, \frac {4 b \left (a +b \right )}{\beta ^{2}}, \coth \left (\frac {\beta x}{2}\right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2}-a \cosh \left (\beta x \right ) y=a b \cosh \left (\beta x \right )-b^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{2}+a \cosh \left (\beta x \right ) y+a b \cosh \left (\beta x \right )-b^{2} \end {array} \]
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: 0.016 (sec). Leaf size: 73
dsolve(diff(y(x),x)=y(x)^2+a*cosh(beta*x)*y(x)+a*b*cosh(beta*x)-b^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {b \left (\int {\mathrm e}^{\frac {-2 b \beta x +\sinh \left (x \beta \right ) a}{\beta }}d x \right )-c_{1} b +{\mathrm e}^{\frac {-2 b \beta x +\sinh \left (x \beta \right ) a}{\beta }}}{-\left (\int {\mathrm e}^{\frac {-2 b \beta x +\sinh \left (x \beta \right ) a}{\beta }}d x \right )+c_{1}} \]
✓ Solution by Mathematica
Time used: 9.815 (sec). Leaf size: 242
DSolve[y'[x]==y[x]^2+a*Cosh[\[Beta]*x]*y[x]+a*b*Cosh[\[Beta]*x]-b^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {b \int _1^{e^{x \beta }}e^{\frac {a \left (K[1]^2-1\right )}{2 \beta K[1]}} K[1]^{-\frac {2 b}{\beta }-1}dK[1]+\beta e^{\frac {a e^{\beta (-x)} \left (e^{2 \beta x}-1\right )}{2 \beta }} \left (e^{\beta x}\right )^{-\frac {2 b}{\beta }}+b c_1}{\int _1^{e^{x \beta }}e^{\frac {a \left (K[1]^2-1\right )}{2 \beta K[1]}} K[1]^{-\frac {2 b}{\beta }-1}dK[1]+c_1} \\ y(x)\to -b \\ y(x)\to -\frac {\beta e^{\frac {a e^{\beta (-x)} \left (e^{2 \beta x}-1\right )}{2 \beta }} \left (e^{\beta x}\right )^{-\frac {2 b}{\beta }}}{\int _1^{e^{x \beta }}e^{\frac {a \left (K[1]^2-1\right )}{2 \beta K[1]}} K[1]^{-\frac {2 b}{\beta }-1}dK[1]}-b \\ \end{align*}