Internal problem ID [10454]
Internal file name [OUTPUT/9402_Monday_June_06_2022_02_23_34_PM_5846101/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: 7.
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]
\[ \boxed {\left (a \sinh \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )=-a \,\lambda ^{2} \sinh \left (\lambda x \right )} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {y^{2} \sinh \left (\lambda x \right ) a -a \,\lambda ^{2} \sinh \left (\lambda x \right )+y^{2} b}{a \sinh \left (\lambda x \right )+b} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -\frac {a \,\lambda ^{2} \sinh \left (\lambda x \right )}{a \sinh \left (\lambda x \right )+b}+\frac {y^{2} \sinh \left (\lambda x \right ) a}{a \sinh \left (\lambda x \right )+b}+\frac {y^{2} b}{a \sinh \left (\lambda x \right )+b} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-\frac {a \,\lambda ^{2} \sinh \left (\lambda x \right )}{a \sinh \left (\lambda x \right )+b}\), \(f_1(x)=0\) 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 &=0\\ f_2^2 f_0 &=-\frac {a \,\lambda ^{2} \sinh \left (\lambda x \right )}{a \sinh \left (\lambda x \right )+b} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} u^{\prime \prime }\left (x \right )-\frac {a \,\lambda ^{2} \sinh \left (\lambda x \right ) u \left (x \right )}{a \sinh \left (\lambda x \right )+b} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = -\cosh \left (\frac {\lambda x}{2}\right ) a c_{1} \left (a^{2}+b^{2}\right )^{\frac {3}{2}} \left (a \sinh \left (\frac {\lambda x}{2}\right )+b \cosh \left (\frac {\lambda x}{2}\right )\right )-2 \left (\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2} c_{1} +\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4} c_{1} -c_{2} \right ) \left (\sinh \left (\frac {\lambda x}{2}\right ) a \cosh \left (\frac {\lambda x}{2}\right )+\frac {b}{2}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = -\frac {4 \left (\left (-\frac {\tanh \left (\frac {\lambda x}{2}\right )^{2} b}{2}+\tanh \left (\frac {\lambda x}{2}\right ) a +\frac {b}{2}\right ) \left (\cosh \left (\frac {\lambda x}{2}\right )^{2}+\sinh \left (\frac {\lambda x}{2}\right )^{2}\right ) \left (\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2} c_{1} +\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4} c_{1} -c_{2} \right ) a \sqrt {a^{2}+b^{2}}+\frac {c_{1} \left (a^{2}+b^{2}\right )^{2} \left (-\frac {b \left (a^{2} \sinh \left (\frac {\lambda x}{2}\right )^{2}+a^{2} \cosh \left (\frac {\lambda x}{2}\right )^{2}-b^{2}\right ) \tanh \left (\frac {\lambda x}{2}\right )^{2}}{2}+a^{2} \left (a \sinh \left (\frac {\lambda x}{2}\right )^{2}+a \cosh \left (\frac {\lambda x}{2}\right )^{2}+2 \sinh \left (\frac {\lambda x}{2}\right ) b \cosh \left (\frac {\lambda x}{2}\right )\right ) \tanh \left (\frac {\lambda x}{2}\right )+\frac {b \left (a^{2} \sinh \left (\frac {\lambda x}{2}\right )^{2}+a^{2} \cosh \left (\frac {\lambda x}{2}\right )^{2}-b^{2}\right )}{2}\right )}{2}\right ) \lambda }{\sqrt {a^{2}+b^{2}}\, \left (-2 \tanh \left (\frac {\lambda x}{2}\right )^{2} b +4 \tanh \left (\frac {\lambda x}{2}\right ) a +2 b \right )} \] Using the above in (1) gives the solution \[ y = \frac {4 \left (\left (-\frac {\tanh \left (\frac {\lambda x}{2}\right )^{2} b}{2}+\tanh \left (\frac {\lambda x}{2}\right ) a +\frac {b}{2}\right ) \left (\cosh \left (\frac {\lambda x}{2}\right )^{2}+\sinh \left (\frac {\lambda x}{2}\right )^{2}\right ) \left (\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2} c_{1} +\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4} c_{1} -c_{2} \right ) a \sqrt {a^{2}+b^{2}}+\frac {c_{1} \left (a^{2}+b^{2}\right )^{2} \left (-\frac {b \left (a^{2} \sinh \left (\frac {\lambda x}{2}\right )^{2}+a^{2} \cosh \left (\frac {\lambda x}{2}\right )^{2}-b^{2}\right ) \tanh \left (\frac {\lambda x}{2}\right )^{2}}{2}+a^{2} \left (a \sinh \left (\frac {\lambda x}{2}\right )^{2}+a \cosh \left (\frac {\lambda x}{2}\right )^{2}+2 \sinh \left (\frac {\lambda x}{2}\right ) b \cosh \left (\frac {\lambda x}{2}\right )\right ) \tanh \left (\frac {\lambda x}{2}\right )+\frac {b \left (a^{2} \sinh \left (\frac {\lambda x}{2}\right )^{2}+a^{2} \cosh \left (\frac {\lambda x}{2}\right )^{2}-b^{2}\right )}{2}\right )}{2}\right ) \lambda }{\sqrt {a^{2}+b^{2}}\, \left (-2 \tanh \left (\frac {\lambda x}{2}\right )^{2} b +4 \tanh \left (\frac {\lambda x}{2}\right ) a +2 b \right ) \left (-\cosh \left (\frac {\lambda x}{2}\right ) a c_{1} \left (a^{2}+b^{2}\right )^{\frac {3}{2}} \left (a \sinh \left (\frac {\lambda x}{2}\right )+b \cosh \left (\frac {\lambda x}{2}\right )\right )-2 \left (\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2} c_{1} +\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4} c_{1} -c_{2} \right ) \left (\sinh \left (\frac {\lambda x}{2}\right ) a \cosh \left (\frac {\lambda x}{2}\right )+\frac {b}{2}\right )\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {4 \left (\left (\cosh \left (\frac {\lambda x}{2}\right )^{2}-\frac {1}{2}\right ) \left (\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2} c_{3} +\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4} c_{3} -1\right ) a \sqrt {a^{2}+b^{2}}+\frac {\left (a^{2} \cosh \left (\frac {\lambda x}{2}\right )^{2}+\sinh \left (\frac {\lambda x}{2}\right ) a b \cosh \left (\frac {\lambda x}{2}\right )-\frac {a^{2}}{2}-\frac {b^{2}}{2}\right ) c_{3} \left (a^{2}+b^{2}\right )^{2}}{2}\right ) \lambda }{\sqrt {a^{2}+b^{2}}\, \left (2 \cosh \left (\frac {\lambda x}{2}\right ) a c_{3} \left (a^{2}+b^{2}\right )^{\frac {3}{2}} \left (a \sinh \left (\frac {\lambda x}{2}\right )+b \cosh \left (\frac {\lambda x}{2}\right )\right )+4 \left (\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2} c_{3} +\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4} c_{3} -1\right ) \left (\sinh \left (\frac {\lambda x}{2}\right ) a \cosh \left (\frac {\lambda x}{2}\right )+\frac {b}{2}\right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {4 \left (\left (\cosh \left (\frac {\lambda x}{2}\right )^{2}-\frac {1}{2}\right ) \left (\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2} c_{3} +\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4} c_{3} -1\right ) a \sqrt {a^{2}+b^{2}}+\frac {\left (a^{2} \cosh \left (\frac {\lambda x}{2}\right )^{2}+\sinh \left (\frac {\lambda x}{2}\right ) a b \cosh \left (\frac {\lambda x}{2}\right )-\frac {a^{2}}{2}-\frac {b^{2}}{2}\right ) c_{3} \left (a^{2}+b^{2}\right )^{2}}{2}\right ) \lambda }{\sqrt {a^{2}+b^{2}}\, \left (2 \cosh \left (\frac {\lambda x}{2}\right ) a c_{3} \left (a^{2}+b^{2}\right )^{\frac {3}{2}} \left (a \sinh \left (\frac {\lambda x}{2}\right )+b \cosh \left (\frac {\lambda x}{2}\right )\right )+4 \left (\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2} c_{3} +\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4} c_{3} -1\right ) \left (\sinh \left (\frac {\lambda x}{2}\right ) a \cosh \left (\frac {\lambda x}{2}\right )+\frac {b}{2}\right )\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {4 \left (\left (\cosh \left (\frac {\lambda x}{2}\right )^{2}-\frac {1}{2}\right ) \left (\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2} c_{3} +\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4} c_{3} -1\right ) a \sqrt {a^{2}+b^{2}}+\frac {\left (a^{2} \cosh \left (\frac {\lambda x}{2}\right )^{2}+\sinh \left (\frac {\lambda x}{2}\right ) a b \cosh \left (\frac {\lambda x}{2}\right )-\frac {a^{2}}{2}-\frac {b^{2}}{2}\right ) c_{3} \left (a^{2}+b^{2}\right )^{2}}{2}\right ) \lambda }{\sqrt {a^{2}+b^{2}}\, \left (2 \cosh \left (\frac {\lambda x}{2}\right ) a c_{3} \left (a^{2}+b^{2}\right )^{\frac {3}{2}} \left (a \sinh \left (\frac {\lambda x}{2}\right )+b \cosh \left (\frac {\lambda x}{2}\right )\right )+4 \left (\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2} c_{3} +\operatorname {arctanh}\left (\frac {-b \tanh \left (\frac {\lambda x}{2}\right )+a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4} c_{3} -1\right ) \left (\sinh \left (\frac {\lambda x}{2}\right ) a \cosh \left (\frac {\lambda x}{2}\right )+\frac {b}{2}\right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (a \sinh \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )=-a \,\lambda ^{2} \sinh \left (\lambda x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{2} \sinh \left (\lambda x \right ) a -a \,\lambda ^{2} \sinh \left (\lambda x \right )+y^{2} b}{a \sinh \left (\lambda x \right )+b} \end {array} \]
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: trying Riccati_symmetries trying Riccati to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = lambda^2*sinh(lambda*x)*a*y(x)/(sinh(lambda*x)*a+b), y(x)` *** Su Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] <- linear_1 successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 250
dsolve((a*sinh(lambda*x)+b)*(diff(y(x),x)-y(x)^2)+a*lambda^2*sinh(lambda*x)=0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {4 \left (\left (\operatorname {arctanh}\left (\frac {-\tanh \left (\frac {x \lambda }{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2}+\operatorname {arctanh}\left (\frac {-\tanh \left (\frac {x \lambda }{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4}-c_{1} \right ) a \left (\cosh \left (\frac {x \lambda }{2}\right )^{2}-\frac {1}{2}\right ) \sqrt {a^{2}+b^{2}}+\frac {\left (a^{2}+b^{2}\right )^{2} \left (a^{2} \cosh \left (\frac {x \lambda }{2}\right )^{2}+a b \cosh \left (\frac {x \lambda }{2}\right ) \sinh \left (\frac {x \lambda }{2}\right )-\frac {a^{2}}{2}-\frac {b^{2}}{2}\right )}{2}\right ) \lambda }{\sqrt {a^{2}+b^{2}}\, \left (2 a \cosh \left (\frac {x \lambda }{2}\right ) \left (a^{2}+b^{2}\right )^{\frac {3}{2}} \left (a \sinh \left (\frac {x \lambda }{2}\right )+b \cosh \left (\frac {x \lambda }{2}\right )\right )+4 \left (\operatorname {arctanh}\left (\frac {-\tanh \left (\frac {x \lambda }{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2}+\operatorname {arctanh}\left (\frac {-\tanh \left (\frac {x \lambda }{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4}-c_{1} \right ) \left (\sinh \left (\frac {x \lambda }{2}\right ) a \cosh \left (\frac {x \lambda }{2}\right )+\frac {b}{2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 24.532 (sec). Leaf size: 202
DSolve[(a*Sinh[\[Lambda]*x]+b)*(y'[x]-y[x]^2)+a*\[Lambda]^2*Sinh[\[Lambda]*x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\lambda \left (\sqrt {-a^2-b^2} (b-a \sinh (\lambda x))+a \cosh (\lambda x) \left (2 b \arctan \left (\frac {a-b \tanh \left (\frac {\lambda x}{2}\right )}{\sqrt {-a^2-b^2}}\right )-c_1 \lambda \left (-a^2-b^2\right )^{3/2}\right )\right )}{-a \sqrt {-a^2-b^2} \cosh (\lambda x)+(a \sinh (\lambda x)+b) \left (2 b \arctan \left (\frac {a-b \tanh \left (\frac {\lambda x}{2}\right )}{\sqrt {-a^2-b^2}}\right )-c_1 \lambda \left (-a^2-b^2\right )^{3/2}\right )} \\ y(x)\to -\frac {a \lambda \cosh (\lambda x)}{a \sinh (\lambda x)+b} \\ \end{align*}