Internal problem ID [10571]
Internal file name [OUTPUT/9519_Monday_June_06_2022_03_03_20_PM_88076586/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: 15.
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]
\[ \boxed {y^{\prime }-\lambda \arccos \left (x \right )^{n} y^{2}=\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arccos \left (x \right )^{n}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \arccos \left (x \right )^{n} \lambda \,y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arccos \left (x \right )^{n} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \arccos \left (x \right )^{n} \lambda \,y^{2}+\frac {\beta m \,x^{m}}{x}-\lambda \,\beta ^{2} x^{2 m} \arccos \left (x \right )^{n} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arccos \left (x \right )^{n}\), \(f_1(x)=0\) and \(f_2(x)=\arccos \left (x \right )^{n} \lambda \). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\arccos \left (x \right )^{n} \lambda 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' &=-\frac {\arccos \left (x \right )^{n} n \lambda }{\sqrt {-x^{2}+1}\, \arccos \left (x \right )}\\ f_1 f_2 &=0\\ f_2^2 f_0 &=\arccos \left (x \right )^{2 n} \lambda ^{2} \left (\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arccos \left (x \right )^{n}\right ) \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \arccos \left (x \right )^{n} \lambda u^{\prime \prime }\left (x \right )+\frac {\arccos \left (x \right )^{n} n \lambda u^{\prime }\left (x \right )}{\sqrt {-x^{2}+1}\, \arccos \left (x \right )}+\arccos \left (x \right )^{2 n} \lambda ^{2} \left (\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arccos \left (x \right )^{n}\right ) u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )+\frac {n \textit {\_Y}^{\prime }\left (x \right )}{\arccos \left (x \right ) \sqrt {-x^{2}+1}}-x^{2 m} \beta ^{2} \textit {\_Y} \left (x \right ) \lambda ^{2} \arccos \left (x \right )^{2 n}+m \beta \,x^{m -1} \lambda \textit {\_Y} \left (x \right ) \arccos \left (x \right )^{n}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )+\frac {n \textit {\_Y}^{\prime }\left (x \right )}{\arccos \left (x \right ) \sqrt {-x^{2}+1}}-x^{2 m} \beta ^{2} \textit {\_Y} \left (x \right ) \lambda ^{2} \arccos \left (x \right )^{2 n}+m \beta \,x^{m -1} \lambda \textit {\_Y} \left (x \right ) \arccos \left (x \right )^{n}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )+\frac {n \textit {\_Y}^{\prime }\left (x \right )}{\arccos \left (x \right ) \sqrt {-x^{2}+1}}-x^{2 m} \beta ^{2} \textit {\_Y} \left (x \right ) \lambda ^{2} \arccos \left (x \right )^{2 n}+m \beta \,x^{m -1} \lambda \textit {\_Y} \left (x \right ) \arccos \left (x \right )^{n}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \arccos \left (x \right )^{-n}}{\lambda \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )+\frac {n \textit {\_Y}^{\prime }\left (x \right )}{\arccos \left (x \right ) \sqrt {-x^{2}+1}}-x^{2 m} \beta ^{2} \textit {\_Y} \left (x \right ) \lambda ^{2} \arccos \left (x \right )^{2 n}+m \beta \,x^{m -1} \lambda \textit {\_Y} \left (x \right ) \arccos \left (x \right )^{n}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\left (-x^{2 m} \arccos \left (x \right )^{1+2 n} \textit {\_Y} \left (x \right ) \beta ^{2} \lambda ^{2}+\arccos \left (x \right )^{n +1} x^{m -1} \textit {\_Y} \left (x \right ) \beta \lambda m +\textit {\_Y}^{\prime \prime }\left (x \right ) \arccos \left (x \right )\right ) \sqrt {-x^{2}+1}+n \textit {\_Y}^{\prime }\left (x \right )}{\sqrt {-x^{2}+1}\, \arccos \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \arccos \left (x \right )^{-n}}{\lambda \operatorname {DESol}\left (\left \{\frac {\left (-x^{1+2 m} \arccos \left (x \right )^{1+2 n} \textit {\_Y} \left (x \right ) \beta ^{2} \lambda ^{2}+\arccos \left (x \right )^{n +1} x^{m} \textit {\_Y} \left (x \right ) \beta \lambda m +\textit {\_Y}^{\prime \prime }\left (x \right ) \arccos \left (x \right ) x \right ) \sqrt {-x^{2}+1}+n \textit {\_Y}^{\prime }\left (x \right ) x}{\sqrt {-x^{2}+1}\, x \arccos \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\left (-x^{2 m} \arccos \left (x \right )^{1+2 n} \textit {\_Y} \left (x \right ) \beta ^{2} \lambda ^{2}+\arccos \left (x \right )^{n +1} x^{m -1} \textit {\_Y} \left (x \right ) \beta \lambda m +\textit {\_Y}^{\prime \prime }\left (x \right ) \arccos \left (x \right )\right ) \sqrt {-x^{2}+1}+n \textit {\_Y}^{\prime }\left (x \right )}{\sqrt {-x^{2}+1}\, \arccos \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \arccos \left (x \right )^{-n}}{\lambda \operatorname {DESol}\left (\left \{\frac {\left (-x^{1+2 m} \arccos \left (x \right )^{1+2 n} \textit {\_Y} \left (x \right ) \beta ^{2} \lambda ^{2}+\arccos \left (x \right )^{n +1} x^{m} \textit {\_Y} \left (x \right ) \beta \lambda m +\textit {\_Y}^{\prime \prime }\left (x \right ) \arccos \left (x \right ) x \right ) \sqrt {-x^{2}+1}+n \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 )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\left (-x^{2 m} \arccos \left (x \right )^{1+2 n} \textit {\_Y} \left (x \right ) \beta ^{2} \lambda ^{2}+\arccos \left (x \right )^{n +1} x^{m -1} \textit {\_Y} \left (x \right ) \beta \lambda m +\textit {\_Y}^{\prime \prime }\left (x \right ) \arccos \left (x \right )\right ) \sqrt {-x^{2}+1}+n \textit {\_Y}^{\prime }\left (x \right )}{\sqrt {-x^{2}+1}\, \arccos \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \arccos \left (x \right )^{-n}}{\lambda \operatorname {DESol}\left (\left \{\frac {\left (-x^{1+2 m} \arccos \left (x \right )^{1+2 n} \textit {\_Y} \left (x \right ) \beta ^{2} \lambda ^{2}+\arccos \left (x \right )^{n +1} x^{m} \textit {\_Y} \left (x \right ) \beta \lambda m +\textit {\_Y}^{\prime \prime }\left (x \right ) \arccos \left (x \right ) x \right ) \sqrt {-x^{2}+1}+n \textit {\_Y}^{\prime }\left (x \right ) x}{\sqrt {-x^{2}+1}\, x \arccos \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\lambda \arccos \left (x \right )^{n} y^{2}=\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arccos \left (x \right )^{n} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\lambda \arccos \left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arccos \left (x \right )^{n} \end {array} \]
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) = -n*(diff(y(x), x))/((-x^2+1)^(1/2)*arccos(x))+arccos(x)^n*lambda*beta* 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 with_periodic_functions in the coefficients -> Trying a change of variables to reduce to Bernoulli -> Calling odsolve with the ODE`, diff(y(x), x)-(arccos(x)^n*lambda*y(x)^2+y(x)+x^2*(beta*m*x^(m-1)-lambda*beta^2*x^(2*m)*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`
✗ Solution by Maple
dsolve(diff(y(x),x)=lambda*arccos(x)^n*y(x)^2+beta*m*x^(m-1)-lambda*beta^2*x^(2*m)*arccos(x)^n,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==\[Lambda]*ArcCos[x]^n*y[x]^2+\[Beta]*m*x^(m-1)-\[Lambda]*\[Beta]^2*x^(2*m)*ArcCos[x]^n,y[x],x,IncludeSingularSolutions -> True]
Not solved