Internal problem ID [10579]
Internal file name [OUTPUT/9527_Monday_June_06_2022_03_04_28_PM_46831198/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 1.2.7-3. Equations containing
arctangent.
Problem number: 23.
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]
\[ \boxed {y^{\prime }-\lambda \arctan \left (x \right )^{n} y^{2}+b \lambda \,x^{m} \arctan \left (x \right )^{n} y=b m \,x^{m -1}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \lambda \arctan \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arctan \left (x \right )^{n} y +b m \,x^{m -1} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \lambda \arctan \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arctan \left (x \right )^{n} y +\frac {b m \,x^{m}}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=b m \,x^{m -1}\), \(f_1(x)=-b \lambda \,x^{m} \arctan \left (x \right )^{n}\) and \(f_2(x)=\arctan \left (x \right )^{n} \lambda \). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\arctan \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 {\arctan \left (x \right )^{n} n \lambda }{\left (x^{2}+1\right ) \arctan \left (x \right )}\\ f_1 f_2 &=-b \,\lambda ^{2} x^{m} \arctan \left (x \right )^{2 n}\\ f_2^2 f_0 &=\arctan \left (x \right )^{2 n} \lambda ^{2} b m \,x^{m -1} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \arctan \left (x \right )^{n} \lambda u^{\prime \prime }\left (x \right )-\left (\frac {\arctan \left (x \right )^{n} n \lambda }{\left (x^{2}+1\right ) \arctan \left (x \right )}-b \,\lambda ^{2} x^{m} \arctan \left (x \right )^{2 n}\right ) u^{\prime }\left (x \right )+\arctan \left (x \right )^{2 n} \lambda ^{2} b m \,x^{m -1} u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {n \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right )}+\arctan \left (x \right )^{n} b \lambda \,x^{m} \textit {\_Y}^{\prime }\left (x \right )+\arctan \left (x \right )^{n} \lambda b m \,x^{m -1} \textit {\_Y} \left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {n \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right )}+\arctan \left (x \right )^{n} b \lambda \,x^{m} \textit {\_Y}^{\prime }\left (x \right )+\arctan \left (x \right )^{n} \lambda b m \,x^{m -1} \textit {\_Y} \left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {n \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right )}+\arctan \left (x \right )^{n} b \lambda \,x^{m} \textit {\_Y}^{\prime }\left (x \right )+\arctan \left (x \right )^{n} \lambda b m \,x^{m -1} \textit {\_Y} \left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \arctan \left (x \right )^{-n}}{\lambda \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {n \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right )}+\arctan \left (x \right )^{n} b \lambda \,x^{m} \textit {\_Y}^{\prime }\left (x \right )+\arctan \left (x \right )^{n} \lambda b m \,x^{m -1} \textit {\_Y} \left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {b \lambda \left (x^{2}+1\right ) \left (m \textit {\_Y} \left (x \right ) x^{m -1}+\textit {\_Y}^{\prime }\left (x \right ) x^{m}\right ) \arctan \left (x \right )^{n +1}+\textit {\_Y}^{\prime \prime }\left (x \right ) \left (x^{2}+1\right ) \arctan \left (x \right )-n \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \arctan \left (x \right )^{-n}}{\lambda \operatorname {DESol}\left (\left \{\frac {b \lambda \left (x^{2+m} m \textit {\_Y} \left (x \right )+m \,x^{m} \textit {\_Y} \left (x \right )+x^{m +3} \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) x^{1+m}\right ) \arctan \left (x \right )^{n +1}-x \left (\left (-x^{2}-1\right ) \arctan \left (x \right ) \textit {\_Y}^{\prime \prime }\left (x \right )+n \textit {\_Y}^{\prime }\left (x \right )\right )}{\arctan \left (x \right ) x \left (x^{2}+1\right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {b \lambda \left (x^{2}+1\right ) \left (m \textit {\_Y} \left (x \right ) x^{m -1}+\textit {\_Y}^{\prime }\left (x \right ) x^{m}\right ) \arctan \left (x \right )^{n +1}+\textit {\_Y}^{\prime \prime }\left (x \right ) \left (x^{2}+1\right ) \arctan \left (x \right )-n \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \arctan \left (x \right )^{-n}}{\lambda \operatorname {DESol}\left (\left \{\frac {b \lambda \left (x^{2+m} m \textit {\_Y} \left (x \right )+m \,x^{m} \textit {\_Y} \left (x \right )+x^{m +3} \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) x^{1+m}\right ) \arctan \left (x \right )^{n +1}-x \left (\left (-x^{2}-1\right ) \arctan \left (x \right ) \textit {\_Y}^{\prime \prime }\left (x \right )+n \textit {\_Y}^{\prime }\left (x \right )\right )}{\arctan \left (x \right ) x \left (x^{2}+1\right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {b \lambda \left (x^{2}+1\right ) \left (m \textit {\_Y} \left (x \right ) x^{m -1}+\textit {\_Y}^{\prime }\left (x \right ) x^{m}\right ) \arctan \left (x \right )^{n +1}+\textit {\_Y}^{\prime \prime }\left (x \right ) \left (x^{2}+1\right ) \arctan \left (x \right )-n \textit {\_Y}^{\prime }\left (x \right )}{\left (x^{2}+1\right ) \arctan \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \arctan \left (x \right )^{-n}}{\lambda \operatorname {DESol}\left (\left \{\frac {b \lambda \left (x^{2+m} m \textit {\_Y} \left (x \right )+m \,x^{m} \textit {\_Y} \left (x \right )+x^{m +3} \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime }\left (x \right ) x^{1+m}\right ) \arctan \left (x \right )^{n +1}-x \left (\left (-x^{2}-1\right ) \arctan \left (x \right ) \textit {\_Y}^{\prime \prime }\left (x \right )+n \textit {\_Y}^{\prime }\left (x \right )\right )}{\arctan \left (x \right ) x \left (x^{2}+1\right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\lambda \arctan \left (x \right )^{n} y^{2}+b \lambda \,x^{m} \arctan \left (x \right )^{n} y=b m \,x^{m -1} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\lambda \arctan \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arctan \left (x \right )^{n} y+b m \,x^{m -1} \end {array} \]
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^m*arctan(x)^n*arctan(x)*b*lambda*x^2+b*lambda*x^m*arctan(x)^n*arct 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 -> 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 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 with_periodic_functions in the coefficients <- 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 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 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 with_periodic_functions in the coefficients <- 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)-(arctan(x)^n*lambda*y(x)^2+y(x)-b*lambda*x^m*arctan(x)^n*y(x)*x+x^2*b*m*x^(m-1))/ 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*arctan(x)^n*y(x)^2-b*lambda*x^m*arctan(x)^n*y(x)+b*m*x^(m-1),y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==\[Lambda]*ArcTan[x]^n*y[x]^2-b*\[Lambda]*x^m*ArcTan[x]^n*y[x]+b*m*x^(m-1),y[x],x,IncludeSingularSolutions -> True]
Not solved