problem 18

Internal problem ID [10574]
Internal ﬁle name [OUTPUT/9522_Monday_June_06_2022_03_03_46_PM_39589339/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 1.2.7-2. Equations containing arccosine.
Problem number: 18.
ODE order: 1.
ODE degree: 1.

\[ \boxed {x y^{\prime }-\left (a \,x^{2 m} y^{2}+b \,x^{n} y+c \right ) \arccos \left (x \right )^{m}+n y=0} \]

15.9.1 Solving as riccati ode

In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {x^{2 m} \arccos \left (x \right )^{m} a \,y^{2}+x^{n} \arccos \left (x \right )^{m} b y +\arccos \left (x \right )^{m} c -n y}{x} \end {align*}

This is a Riccati ODE. Comparing the ODE to solve \[ y' = \frac {x^{2 m} \arccos \left (x \right )^{m} a \,y^{2}}{x}+\frac {x^{n} \arccos \left (x \right )^{m} b y}{x}+\frac {\arccos \left (x \right )^{m} c}{x}-\frac {n y}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {\arccos \left (x \right )^{m} c}{x}\), \(f_1(x)=\frac {\arccos \left (x \right )^{m} x^{n} b -n}{x}\) and \(f_2(x)=\frac {x^{2 m} \arccos \left (x \right )^{m} a}{x}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\frac {x^{2 m} \arccos \left (x \right )^{m} a u}{x}} \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 {2 x^{2 m} m \arccos \left (x \right )^{m} a}{x^{2}}-\frac {x^{2 m} \arccos \left (x \right )^{m} m a}{\sqrt {-x^{2}+1}\, \arccos \left (x \right ) x}-\frac {x^{2 m} \arccos \left (x \right )^{m} a}{x^{2}}\\ f_1 f_2 &=\frac {\left (\arccos \left (x \right )^{m} x^{n} b -n \right ) x^{2 m} \arccos \left (x \right )^{m} a}{x^{2}}\\ f_2^2 f_0 &=\frac {x^{4 m} \arccos \left (x \right )^{3 m} a^{2} c}{x^{3}} \end {align*}

Substituting the above terms back in equation (2) gives \begin {align*} \frac {x^{2 m} \arccos \left (x \right )^{m} a u^{\prime \prime }\left (x \right )}{x}-\left (\frac {2 x^{2 m} m \arccos \left (x \right )^{m} a}{x^{2}}-\frac {x^{2 m} \arccos \left (x \right )^{m} m a}{\sqrt {-x^{2}+1}\, \arccos \left (x \right ) x}-\frac {x^{2 m} \arccos \left (x \right )^{m} a}{x^{2}}+\frac {\left (\arccos \left (x \right )^{m} x^{n} b -n \right ) x^{2 m} \arccos \left (x \right )^{m} a}{x^{2}}\right ) u^{\prime }\left (x \right )+\frac {x^{4 m} \arccos \left (x \right )^{3 m} a^{2} c u \left (x \right )}{x^{3}} &=0 \end {align*}

\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {2 m \textit {\_Y}^{\prime }\left (x \right )}{x}+\frac {m \textit {\_Y}^{\prime }\left (x \right )}{\arccos \left (x \right ) \sqrt {-x^{2}+1}}+\frac {\textit {\_Y}^{\prime }\left (x \right )}{x}-b \,x^{n -1} \arccos \left (x \right )^{m} \textit {\_Y}^{\prime }\left (x \right )+\frac {n \textit {\_Y}^{\prime }\left (x \right )}{x}+a c \,x^{2 m -2} \textit {\_Y} \left (x \right ) \arccos \left (x \right )^{2 m}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {2 m \textit {\_Y}^{\prime }\left (x \right )}{x}+\frac {m \textit {\_Y}^{\prime }\left (x \right )}{\arccos \left (x \right ) \sqrt {-x^{2}+1}}+\frac {\textit {\_Y}^{\prime }\left (x \right )}{x}-b \,x^{n -1} \arccos \left (x \right )^{m} \textit {\_Y}^{\prime }\left (x \right )+\frac {n \textit {\_Y}^{\prime }\left (x \right )}{x}+a c \,x^{2 m -2} \textit {\_Y} \left (x \right ) \arccos \left (x \right )^{2 m}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {2 m \textit {\_Y}^{\prime }\left (x \right )}{x}+\frac {m \textit {\_Y}^{\prime }\left (x \right )}{\arccos \left (x \right ) \sqrt {-x^{2}+1}}+\frac {\textit {\_Y}^{\prime }\left (x \right )}{x}-b \,x^{n -1} \arccos \left (x \right )^{m} \textit {\_Y}^{\prime }\left (x \right )+\frac {n \textit {\_Y}^{\prime }\left (x \right )}{x}+a c \,x^{2 m -2} \textit {\_Y} \left (x \right ) \arccos \left (x \right )^{2 m}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-2 m} \arccos \left (x \right )^{-m} x}{a \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {2 m \textit {\_Y}^{\prime }\left (x \right )}{x}+\frac {m \textit {\_Y}^{\prime }\left (x \right )}{\arccos \left (x \right ) \sqrt {-x^{2}+1}}+\frac {\textit {\_Y}^{\prime }\left (x \right )}{x}-b \,x^{n -1} \arccos \left (x \right )^{m} \textit {\_Y}^{\prime }\left (x \right )+\frac {n \textit {\_Y}^{\prime }\left (x \right )}{x}+a c \,x^{2 m -2} \textit {\_Y} \left (x \right ) \arccos \left (x \right )^{2 m}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution

\[ y = -\frac {x^{-2 m +1} \arccos \left (x \right )^{-m} \left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\left (x^{2 m -1} \arccos \left (x \right )^{1+2 m} \textit {\_Y} \left (x \right ) a c -\textit {\_Y}^{\prime }\left (x \right ) \arccos \left (x \right )^{1+m} x^{n} b -2 \left (-\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) x}{2}+\textit {\_Y}^{\prime }\left (x \right ) \left (m -\frac {n}{2}-\frac {1}{2}\right )\right ) \arccos \left (x \right )\right ) \sqrt {-x^{2}+1}+m \textit {\_Y}^{\prime }\left (x \right ) x}{\sqrt {-x^{2}+1}\, x \arccos \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right )}{a \operatorname {DESol}\left (\left \{\frac {\left (\textit {\_Y} \left (x \right ) x^{2 m} \arccos \left (x \right )^{1+2 m} a c -\textit {\_Y}^{\prime }\left (x \right ) x^{n +1} \arccos \left (x \right )^{1+m} b -2 x \left (-\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) x}{2}+\textit {\_Y}^{\prime }\left (x \right ) \left (m -\frac {n}{2}-\frac {1}{2}\right )\right ) \arccos \left (x \right )\right ) \sqrt {-x^{2}+1}+m \textit {\_Y}^{\prime }\left (x \right ) x^{2}}{\sqrt {-x^{2}+1}\, \arccos \left (x \right ) x^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {x^{-2 m +1} \arccos \left (x \right )^{-m} \left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\left (x^{2 m -1} \arccos \left (x \right )^{1+2 m} \textit {\_Y} \left (x \right ) a c -\textit {\_Y}^{\prime }\left (x \right ) \arccos \left (x \right )^{1+m} x^{n} b -2 \left (-\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) x}{2}+\textit {\_Y}^{\prime }\left (x \right ) \left (m -\frac {n}{2}-\frac {1}{2}\right )\right ) \arccos \left (x \right )\right ) \sqrt {-x^{2}+1}+m \textit {\_Y}^{\prime }\left (x \right ) x}{\sqrt {-x^{2}+1}\, x \arccos \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right )}{a \operatorname {DESol}\left (\left \{\frac {\left (\textit {\_Y} \left (x \right ) x^{2 m} \arccos \left (x \right )^{1+2 m} a c -\textit {\_Y}^{\prime }\left (x \right ) x^{n +1} \arccos \left (x \right )^{1+m} b -2 x \left (-\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) x}{2}+\textit {\_Y}^{\prime }\left (x \right ) \left (m -\frac {n}{2}-\frac {1}{2}\right )\right ) \arccos \left (x \right )\right ) \sqrt {-x^{2}+1}+m \textit {\_Y}^{\prime }\left (x \right ) x^{2}}{\sqrt {-x^{2}+1}\, \arccos \left (x \right ) x^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \\ \end{align*}

Veriﬁcation of solutions

15.9.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y^{\prime }-\left (a \,x^{2 m} y^{2}+b \,x^{n} y+c \right ) \arccos \left (x \right )^{m}+n y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\left (a \,x^{2 m} y^{2}+b \,x^{n} y+c \right ) \arccos \left (x \right )^{m}-n y}{x} \end {array} \]

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) = (x^(n-1)*(-x^2+1)^(1/2)*arccos(x)*arccos(x)^m*b*x+2*(-x^2+1)^(1/2)*arc 
      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 
      <- unable to find a useful change of variables 
         trying a symmetry of the form [xi=0, eta=F(x)] 
         trying 2nd order exact linear 
         trying symmetries linear in x and y(x) 
         trying to convert to a linear ODE with constant coefficients 
         trying 2nd order, integrating factor of the form mu(x,y) 
         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 
         <- unable to find a useful change of variables 
            trying a symmetry of the form [xi=0, eta=F(x)] 
         trying to convert to an ODE of Bessel type 
   -> Trying a change of variables to reduce to Bernoulli 
   -> Calling odsolve with the ODE`, diff(y(x), x)-(x^(-1+2*m)*a*arccos(x)^m*y(x)^2+y(x)+(x^(n-1)*arccos(x)^m*b-n/x)*y(x)*x+x*arccos 
      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 inverse_Riccati 
      trying 1st order ODE linearizable_by_differentiation 
   -> trying a symmetry pattern of the form [F(x)*G(y), 0] 
   -> trying a symmetry pattern of the form [0, F(x)*G(y)] 
   -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] 
trying inverse_Riccati 
trying 1st order ODE linearizable_by_differentiation 
--- Trying Lie symmetry methods, 1st order --- 
`, `-> Computing symmetries using: way = 4 
`, `-> Computing symmetries using: way = 2 
`, `-> Computing symmetries using: way = 6`

15.9 problem 18

15.9.1 Solving as riccati ode

15.9.2 Maple step by step solution