Internal problem ID [10461]
Internal file name [OUTPUT/9409_Monday_June_06_2022_02_27_10_PM_68507380/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: 14.
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]
\[ \boxed {y^{\prime }-y^{2} \sinh \left (\lambda x \right ) a=b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{n}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= y^{2} \sinh \left (\lambda x \right ) a +b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{n} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = y^{2} \sinh \left (\lambda x \right ) a +b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{n} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{n}\), \(f_1(x)=0\) and \(f_2(x)=a \sinh \left (\lambda x \right )\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{a \sinh \left (\lambda x \right ) 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' &=a \lambda \cosh \left (\lambda x \right )\\ f_1 f_2 &=0\\ f_2^2 f_0 &=a^{2} \sinh \left (\lambda x \right )^{3} b \cosh \left (\lambda x \right )^{n} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} a \sinh \left (\lambda x \right ) u^{\prime \prime }\left (x \right )-a \lambda \cosh \left (\lambda x \right ) u^{\prime }\left (x \right )+a^{2} \sinh \left (\lambda x \right )^{3} b \cosh \left (\lambda x \right )^{n} u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \sqrt {\cosh \left (\lambda x \right )}\, \left (c_{1} \operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )+c_{2} \operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = -\frac {\sinh \left (\lambda x \right ) \left (\sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}} \operatorname {BesselJ}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right ) c_{1} +\sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}} \operatorname {BesselY}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right ) c_{2} -\lambda \left (c_{1} \operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )+c_{2} \operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )\right )\right )}{\sqrt {\cosh \left (\lambda x \right )}} \] Using the above in (1) gives the solution \[ y = \frac {\sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}} \operatorname {BesselJ}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right ) c_{1} +\sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}} \operatorname {BesselY}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right ) c_{2} -\lambda \left (c_{1} \operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )+c_{2} \operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )\right )}{\cosh \left (\lambda x \right ) a \left (c_{1} \operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )+c_{2} \operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {\left (-\sqrt {b}\, \left (\operatorname {BesselJ}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right ) c_{3} +\operatorname {BesselY}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )\right ) \cosh \left (\lambda x \right )^{1+\frac {n}{2}} \sqrt {a}+\lambda \left (c_{3} \operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )+\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )\right )\right ) \operatorname {sech}\left (\lambda x \right )}{\left (c_{3} \operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )+\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )\right ) a} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\left (-\sqrt {b}\, \left (\operatorname {BesselJ}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right ) c_{3} +\operatorname {BesselY}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )\right ) \cosh \left (\lambda x \right )^{1+\frac {n}{2}} \sqrt {a}+\lambda \left (c_{3} \operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )+\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )\right )\right ) \operatorname {sech}\left (\lambda x \right )}{\left (c_{3} \operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )+\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )\right ) a} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\left (-\sqrt {b}\, \left (\operatorname {BesselJ}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right ) c_{3} +\operatorname {BesselY}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )\right ) \cosh \left (\lambda x \right )^{1+\frac {n}{2}} \sqrt {a}+\lambda \left (c_{3} \operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )+\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )\right )\right ) \operatorname {sech}\left (\lambda x \right )}{\left (c_{3} \operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )+\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (\lambda x \right )^{1+\frac {n}{2}}}{\lambda \left (2+n \right )}\right )\right ) a} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2} \sinh \left (\lambda x \right ) a =b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{n} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{2} \sinh \left (\lambda x \right ) a +b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{n} \end {array} \]
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: trying Riccati_symmetries trying Riccati to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = lambda*cosh(lambda*x)*(diff(y(x), x))/sinh(lambda*x)-sinh(lambda*x)^2* Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying an equivalence, under non-integer power transformations, to LODEs admitting Liouvillian solutions. -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel <- Bessel successful <- special function solution successful Change of variables used: [x = arccosh(t)/lambda] Linear ODE actually solved: 4*(t-1)^(1/2)*(t+1)^(1/2)*t^n*a*b*(t^2-1)*u(t)+4*(t-1)^(1/2)*(t+1)^(1/2)*lambda^2*(t^2-1)*diff(diff(u(t),t),t) = 0 <- change of variables successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 245
dsolve(diff(y(x),x)=a*sinh(lambda*x)*y(x)^2+b*sinh(lambda*x)*cosh(lambda*x)^n,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\operatorname {sech}\left (x \lambda \right ) \left (-\lambda \sqrt {a}\, \left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (x \lambda \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (x \lambda \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )\right )+\left (\operatorname {BesselY}\left (\frac {3+n}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (x \lambda \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {3+n}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (x \lambda \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )\right ) a \sqrt {b}\, \cosh \left (x \lambda \right )^{\frac {n}{2}+1}\right )}{a^{\frac {3}{2}} \left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (x \lambda \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, \cosh \left (x \lambda \right )^{\frac {n}{2}+1}}{\lambda \left (n +2\right )}\right )\right )} \]
✓ Solution by Mathematica
Time used: 1.376 (sec). Leaf size: 667
DSolve[y'[x]==a*Sinh[\[Lambda]*x]*y[x]^2+b*Sinh[\[Lambda]*x]*Cosh[\[Lambda]*x]^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {a} \sqrt {b} c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \cosh ^{\frac {n}{2}}(\lambda x) \operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )-\text {sech}(\lambda x) \left (\operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \left (\sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(\lambda x) \left (\operatorname {BesselJ}\left (\frac {1}{n+2}-1,\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )-\operatorname {BesselJ}\left (1+\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )+\lambda \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )+\sqrt {a} \sqrt {b} c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \cosh ^{\frac {n}{2}+1}(\lambda x) \operatorname {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )+c_1 \lambda \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )}{2 a \left (\operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )+c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )} \\ y(x)\to \frac {\frac {\sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}}(\lambda x) \left (\operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )-\operatorname {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )\right )}{\operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} \cosh ^{\frac {n}{2}+1}(x \lambda )}{n \lambda +2 \lambda }\right )}-\lambda \text {sech}(\lambda x)}{2 a} \\ \end{align*}